'  1  HE 


MEASURE 


STEVEN 


IRLF 


THEORY  OF  MEASUREMENTS 

A  MANUAL  FOR 

PHYSICS  STUDENTS 


BY 

JAMES   S.    STEVENS 

Professor  of  Physics  in  tlie  University  of  Maine 


ILLUSTRATED 


NEW  YORK 

D.  VAN  NOSTRAND  COMPANY 

25  PARK  PLACE 
1915 


7 


COPYRIGHT,  1915 

BY 
D.  VAN  NOSTRAND  COMPANY 


THE  SCIENTIFIC  PRESS 

ROBERT  DRUMMOND  AND  COMPANY 

BROOKLYN.  N.  Y. 


PREFACE 


THIS  book  is  designed  to  be  used  in  either  of  two  ways: 

1.  As  a  Text-book.    The  work  outlined  would  require 
two  semester  hours  for  its  completion.     By  extending  the 
discussions  and  problems,  it  may  be  made  to  cover  three 
semester  hours;    or  by  omitting  portions  of  the  theory, 
the  student  may  gain  a  working  knowledge  of  the  subject 
in   a   shorter  time.     A   "  rule   of  thumb "   knowledge   of 
adjusting  observations,  however,  is  not  to  be  recommended. 

2.  As  a  Laboratory  Guide.    The  work  would    cover  a 
three  years'   course  in  the   physical   laboratory.     During 
the  first  year,  the  student  would  make  use  of  those  portions 
which    are   devoted   to   methods   of   estimating  precision, 
and  the  propagation  of  errors;    in  the  second  year  the 
methods  of  adjustment  of  observations  would  be  used; 
and  in  the  third  year  the  student  should  be  prepared  to 
discuss  his  results  by  the  use  of  empirical  formulae  and 
curves.     The  work  of  the  second  year  is  well  adapted  to 
students  in  junior  courses  in  engineering,  the  adjustment 

of  data  obtained  from  surveys  being  especially  appropriate. 

iii 

345123 


iv  PREFACE 

The  use  of  the  graphic  method  would  be  illustrated  through- 
out the  entire  course. 

Considerable  space  is  devoted  to  the  theory  of  prob- 
ability. This  subject  is  a  fascinating  one  to  students, 
presumably  on  account  of  its  human  interest. 

J.  S.  S. 
UNIVERSITY  OF  MAINE,  ORONO,  ME., 

January,  1915. 


CONTENTS 


PREFACE iii 

NOTATION  .  .  vii 


CHAPTER  I 

INTRODUCTION 1 

MEASUREMENTS 2 

ERRORS 3 

CHAPTER  II 

PROBABILITY 4 

THE  CURVE 9 

THE  INTEGRAL 14 

THE  ARITHMETICAL  MEAN 17 

A  CONSTANT  INTERVAL 17 

WEIGHTS 18 

CHAPTER  III 

THE  ADJUSTMENT  OF  OBSERVATIONS 22 

SHORT  METHODS 32 

ILLUSTRATIONS 32 

CHAPTER  IV 

THE  PRECISION  OF  MEASUREMENTS 36 

GRAPHIC  REPRESENTATION 36 

v 


vi  CONTENTS 

PAGE 

PRECISION  MEASURES  AND  THEIR  DERIVATION 37 

AVERAGE  DEVIATION 38 

PROBABLE  ERROR 38 

CHAPTER  V 

THE  PROPAGATION  OP  ERRORS 44 

THE  DIRECT  PROBLEM 45 

THE  CONVERSE  PROBLEM 49 

FRACTIONAL  METHOD 50 

BEST  MAGNITUDES  AND  RATIOS 51 

CHAPTER  VI 

PLOTTING 54 

METHOD  ILLUSTRATED 55 

CONSTRUCTION  AND  INTERPRETATION  OF  CURVES 57 

APPLICATION  TO  LABORATORY  PROBLEMS 57 

CHAPTER  VII 

NEGLIGIBILITY 62 

APPLICATIONS 63 

CRITERIA 66 

SIGNIFICANT  FIGURES .  69 


,  CHAPTER  VIII 

EMPIRICAL  FORMULAE  AND  CONSTANTS 72 

METHOD  OF  PROCEDURE 72 

TYPES  OF  CURVES 75 

ILLUSTRATIONS 76 


NOTATION 


The  following  notation  will  be  used  in  this  book: 

y  »  Simple  probability; 

P= Compound  probability; 
'  Q  =  Negative  probability; 
m= Single  observation; 
M  =  Mean  of  observations; 

2= True  value  of  a  component  observation; 

£=The  resultant  true  value; 

x= Error; 

0= Residual; 

ft —Measure  of  precision; 

&= Probability  of  error  zero; 

r= Probable  error  of  a  single  observation; 
fo= Probable  error  of  the  mean; 

6= Error  in  a  component  measurement; 

A = Error  in  a  result; 

o.d.= Average  deviation  of  a  single  observation; 
A.D.-  Average  deviation  of  the  mean; 

p= Weight  (Latin  pondus)', 

E= Huge  error. 

vii 


THEORY  OF  MEASUREMENTS 


CHAPTER  I 
INTRODUCTION 

LOBD  KELVIN  has  told  us  that  one's  knowledge  of 
science  begins  when  he  can  measure  what  he  is  speaking 
about  and  express  it  in  numbers.  Every  year  a  vast  num- 
ber of  measurements  are  made  in  physical,  chemical,  and 
engineering  laboratories,  as  well  as  in  laboratories  for 
advanced  research.  We  are  unable,  however,  to  state 
concerning  any  one  of  these  measurements  that  the  result 
is  absolutely  correct.  One  of  the  most  precise  measure- 
ments in  physical  science  is  that  of  the  wave-length  of 
light.  The  wave-length  of  cadmium  light,  measured  by 
a  Michelson  interferometer  and  a  Rowland  grating,  was 
found  to  be 

Ac  =  0.000064384722  cm.  (Michelson) 
^=0.00006438680  cm.  (Rowland). 

Or  we  may  put  it  in  another  way, 

1  meter  contains  1553163.6  wave-lengths  (Michelson), 
1  meter  contains  1553164.1  wave-lengths  (Fabry  and 
Perot). 


2  THEORY  OF  MEASUREMENTS 

These  measurements  were  made  by  different  observers 
using  different  methods.  They  are  remarkable  for  their 
agreement  and  they  give  us  the  wave-length  of  light  with 
sufficient  accuracy  for  all  purposes.  But  they  are  not 
correct,  and  it  is  not  at  all  likely  that  we  shall  ever  know 
the  true  length  of  a  wave  of  cadmium  light. 

It  is  of  great  importance,  however,  that  we  should  be 
able  to  pass  judgment  upon  the  accuracy  of  measurements 
like  these.  To  borrow  another  illustration  from  the  domain 
of  optics,  we  may  assume  the  value  for  the  velocity  of  light 
to  be  299860±30  km.  per  second.  Here  the  number  30 
is  a  measure  of  precision,  which,  if  omitted,  would  make 
the  statement  ridiculous,  since  the  velocity  of  light  has 
not  been  measured  with  sufficient  accuracy  to  be  regarded 
as  correct  to  a  single  kilometer. 

A  student  who  brings  in  a  value  for  g,  resulting  from 
measurements  with  a  simple  pendulum,  of  982.436  centi- 
meters per  second  per  second,  is  likely  to  look  upon  the 
result  with  complacency  until  he  is  made  to  see  that  all 
figures  after  the  8  are  useless  because  erroneous. 

MEASUREMENTS 

Measurements  are  usually  classified  as  follows: 

1.  Direct — when,  for  example,  a  distance  is  measured 
with  a  tape  line. 

2.  Indirect — when  the  density  of  a  cylinder  is  deter- 
mined by  measurements  of  its  length,  diameter,  and  mass. 

3.  Conditioned — when  the  third  angle  of  a  triangle  is 
restricted  by  the  values  of  the  other  two  angles. 

Measurements  not  so  conditioned  are  called  independent. 


INTRODUCTION  3 

ERRORS 
Errors  in  measurement  may  be  divided  into  two  general 


1.  Those  which  may  be  eliminated,  in  part  at  least, 
by  improving  adjustments  and  taking  greater  care  in  the 
method  employed. 

These  include  errors  in  instruments,  such,  for  example, 
as  are  caused  by  faulty  measuring  sticks,  imperfectly 
graduated  circles,  and  poorly  adjusted  balance  beams. 
The  obvious  method  of  correcting  these  errors  is  to  sub- 
stitute good  instruments  for  poor  ones;  or,  when  possible, 
to  eliminate  the  errors  by  compensation. 

Personal  Errors.  These  are  errors  characteristic  of 
the  individual  observer.  If  we  swing  a  pendulum  before 
a  class  of  students  and  ask  them  to  indicate  the  times 
of  greatest  displacement  by  tapping  with  their  pencils, 
the  result  will  well  illustrate  the  personal  equation.  In 
the  experiment  in  wireless  telegraphy  by  which  time 
messages  were  sent  from  Paris  to  Arlington,  each  observer 
was  carefully  rated  in  order  that  this  error  should  be 
guarded  against. 

Mistakes  are,  unfortunately,  too  common  in  students' 
laboratory  work  to  call  for  an  extended  explanation. 
Errors  in  reading  scales,  in  computation,  and  in  tabulation 
may  be  classed  as  mistakes,  and  these  should  become 
less  and  less  frequent  as  the  student  gains  experience. 

2.  The  second  class  consists  of  errors  which  are  inde- 
terminate in  their  nature,  and  may  not  be  entirely  elimi- 
nated, however  much  care  we  may  take  in  our  measure- 
ments, or  by  the  use  of  the  highest  grade  of  apparatus. 
It  is  these  errors  with  which  we  are  concerned  in  the  dis- 
cussions in  this  book. 


CHAPTER  II 
PROBABILITY 

PROFESSOR  JEVONS  in  his  Principles  of  Science  states 
a  truism  which  has  an  important  bearing  upon  the  theory 
of  probability.  "  Perfect  knowledge  alone  can  give  cer- 
tainty, and  in  Nature  perfect  knowledge  would  be  infinite 
knowledge,  which  is  clearly  beyond  our  capacities.  We 
have,  therefore,  to  content  ourselves  with  partial  knowl- 
edge—knowledge mingled  with  ignorance  producing  doubt." 

We  may  interpret  this  to  mean  that  from  the  point 
of  view  of  Omniscience  everything  exists  as  certainty. 
The  path  of  a  leaf  falling  from  a  tree,  as  well  as  that  of  a 
mote  dancing  in  a  sunbeam  is  known  with  as  great  a  cer- 
tainty as  that  of  a  heavy  body  dropped  to  the  earth.  For 
finite  minds,  however,  where  certainty  is  impossible,  the 
ability  to  pass  upon  the  probability  that  an  event  will 
happen  in  a  certain  way  is  the  best  substitute  attainable. 

What  is  highly  probable  in  minds  of  a  certain  order 
of  intelligence,  may  be  improbable  to  others.  To  a  trained 
meteorological  observer,  it  may  appear  extremely  prob- 
able that  it  will  rain  to-morrow;  to  one  who  bases  his 
weather  predictions  upon  popular  superstition,  it  may 
be  improbable.  Again  it  may  seem  probable  to  certain 
people  that  spirit  hands  produced  music  at  a  stance, 
while  to  others  the  probability  becomes  negative. 

4 


PROBABILITY  5 

We  may  look  at  the  subject  in  another  way.  When  a 
championship  game  of  baseball  has  been  finished,  the  result 
becomes  a  certainty  in  the  minds  of  the  spectators;  while 
in  various  parts  of  the  country,  where  the  result  has  not 
as  yet  been  reported,  bets  continue  to  be  made  expressing 
the  probability  of  what  to  other  minds  is  a  certainty. 

If  a  coin  is  tossed  up  under  normal  conditions,  the 
probability  that  it  will  fall  "  heads  "  is  one  out  of  two, 
or  one-half.  This  is  also  the  probability  that  it  will  fall 
"  tails."  We  have,  then,  from  our  notation,  y  =  %,  or,  since 
we  shall  deal  with  more  than  one  event,  P  =  \  and  Q  —  J. 
Since  the  coin  must  fall  in  one  way  or  the  other,  we  have 

P+Q=  1  (the  symbol  for  certainty). 
If  n  coins  are  thrown,  we  have,  by  the  binomial  theorem 


The  first  term  expresses  the  probability  that  all  will 
come  down  heads,  the  second  that  all  but  one  will  come 
down  heads,  etc. 

If  we  take  n  =  6,  the  chances  that  all  will  be  heads 
may  be  expressed  by  the  fraction  ^j,  and  that  five  will 
be  heads  by  /c. 

PROBLEMS 

1.  By  expanding  the  binomial  to  the  proper  number 
of  terms,  show  that  the  chances  for  four,  three,  two,  one, 
and  no  heads  may  be  expressed  by  £f  ,  If  >  if  >  A> 


6  THEORY  OF  MEASUREMENTS 

2.  By  using  an  additional  term  prove  that  it  is  impossible 
to  throw  seven  heads  with  six  coins. 

3.  Plot  a  curve  with  the  data  obtained,  and  preserve 
it  for  future  use. 

4.  If  the  class  is  sufficiently  large  (say  20)  it  will  prove 
an  interesting  exercise  to  take  the  mean  results  of  64  trials 
by  each  student  and  compare  with  the  results  obtained  by 
theory. 

A  better  acquaintance  with  the  laws  of  probability 
may  be  obtained  by  putting  aside  the  formula  and  solving 
the  following  problems  by  an  appeal  to  reason. 

PROBLEMS 

1.  If  two  dice  are  thrown  what  is  the  probability  that 
the  sum  of  the  numbers  will  be  five?    We  first  determine 
the  possible  results  with  two  dice,  which  may  be  obtained 
by  considering  that  each  number  on  one  die  may  appear 
with  each  number  on  the  other.     This  gives  us  6X6  =  36. 
Five  may  be  obtained  either  by  a  four  and  a  one,  or  a  two 
and  a  three.     Now  the  four  may  be  on  the  first  die  and  the 
one  on  the  second,  or  the  one  may  be  on  the  first  and  the 
four  on  the  second.     The  same  is  true  for  the  two  and 
three.     Thus   we   have  four   possibilities,   giving   a   prob- 
ability of  ^. 

2.  What  is  the  probability  of  throwing  one  ace  with 
a  single  die  in  one  throw? 

3.  What  is  the  probability  of  throwing  no  ace  with  a 
single  die  in  one  throw? 

4.  What  is  the  probability  of  throwing  one  ace  in  two 
trials? 


PROBABILITY  7 

5.  What  is  the  probability  of  throwing  two  aces  in  two 
trials? 

6.  What  is  the  probability  of  throwing  no  ace  in  two 
trials? 

7.  What  is  the  probability  of  throwing  only  one  ace 
in  two  trials? 

8.  A  bag  contains  eight  red,  six  black,  and  five  green 
balls.     What  is  the  probability  of  drawing  first  a  red  and 
then  a  black  ball  in  two  trials? 

9.  The  class  record  shows  that  each  student  in  a  class  of 
twenty-five  usually  solves  one  problem  out  of  three  assigned. 
What  is  the  probability  that  an  assigned  problem  will 
be  solved? 

10.  In  Merriman's  Least  Squares  we  have  this  problem: 
Let  a  hundred  coins  be  thrown  up  each  second  by  each 
of  the  inhabitants  of  the  earth.     How  often  will  a  hundred 
heads  be  thrown  in  a  million  years?     (It  will  prove  inter- 
esting for  the  student  to  guess  the  answer  to  this  problem 
before   solving   it.     The   number   of   inhabitants   may   be 
taken  as  one  and  one-half  billion.) 


Before  leaving  the  subject  of  probability  a  few  general 
considerations  may  be  suggested.  It  is  a  rather  prevalent 
notion  that  antecedent  happenings  have  an  effect  upon 
present  probability.  If  a  coin  has  come  down  heads  five 
times  in  succession,  it  is  pretty  difficult  to  convince  the 
average  man  that  the  chances  for  tails  on  the  sixth  throw 
are  not  greater  than  the  chances  for  heads.  Of  course, 
if  the  conditions  are  normal,  the  probability  of  throwing 
heads  is  just  one-half.  (The  derivation  of  the  word  chance 
is  an  interesting  one.)  Such  expressions  as  "  the  turning 


8  THEORY  OF  MEASUREMENTS 

of  luck  "  indicate  the  strong  hold  this  feeling  has  upon 
the  majority  of  people. 

Again,  we  should  not  become  over-confident  from  the 
results  of  the  laws  of  probability.  Interesting  applica- 
tions may  be  found  in  connection  with  the  periodic  law 
of  Mendelee*ff;  the  law  of  Prout,  which  states  that  the 
atomic  weights  of  the  other  elements  are  exact  multiples 
of  that  of  hydrogen;  and  the  kinetic  theory  of  gases  as 
developed  by  Clausius  and  Mayer.  A  case  in  which  the 
probability  almost  becomes  a  certainty  is  illustrated  by 
the  coincidence  of  the  seventy  spectral  lines  in  iron  vapor 
with  those  in  solar  light.  The  possible  arrangement  of  the 
seventy  lines  would  be 

70X69X68X  .  .  .  4X3X2X1. 

But  as  all  possible  arrangements  would  not  apply,  the 
probability  of  a  chance  coincidence  has  been  estimated  by 
Kirchhoff  as  one  in  one  trillion.  Do  we  know  that  iron 
exists  in  the  sun? 

The  following  illustration  may  serve  to  weaken  our 
feeling  of  certainty:  Imagine  a  fly  watching  two  coin- 
cidence pendulums  which  come  together  on  the  eighty- 
first  swing.  While  we  are  about  it,  let  us  imagine  that  in 
the  mind  of  a  fly,  one  second  represents  a  year.  The 
fly  will  watch  the  vibrations  for  ten  seconds  (years)  and 
report  to  another  fly  which  has  just  come  up,  that  quite 
an  extended  series  of  observations  fails  to  discover  the  two 
pendulums  in  coincidence.  The  pendulums  are  watched 
through  fifty,  sixty,  and  seventy  swings  and  it  becomes  a 


PROBABILITY  9 

law  in  flyland  that  pendulums  do  not  get  together.  When 
the  law  has  become  established  by  observations  extending 
over  eighty  fly-years,  the  bell  rings  and  the  assumption 
based  upon  a  very  reasonable  law  of  probability  breaks 
down. 

The  Probability  Curve.  Referring  to  problem  3,  page 
6,  it  will  be  seen  that  the  curve  takes  the  form  indicated 
in  Fig.  1. 

Any  similar  data  from  experiments  which  follow  the 
laws  of  probability  would  yield  such  a  curve  as  this. 


FIG.  1. 

A  familiar  illustration  is  afforded  by  target  practice. 
If  the  target  is  marked  off  into  divisions,  the  distances  of 
these  divisions  from  the  center  may  be  taken  to  represent 
the  magnitude  of  the  errors.  If  a  series  of  shots  be  fired 
by  expert  marksmen  under  normal  conditions,  they  will 
form  the  basis  for  a  curve  which  will  resemble  Fig.  1. 

Since  the  errors  are  plotted  along  the  x-axis  and  their 
corresponding  probabilities  along  the  t/-axis;  and  since 
the  curve  is  seen  to  be  symmetrical  with  respect  to  the 


10 


THEORY  OF  MEASUREMENTS 


2/-axis,  and  the  z-axis  is  an  asymtote,  it  will  be  seen  that  its 
equation  must  take  some  such  form  as 


A  rigid  deduction  of  this  probability  equation  has  been 
given  by  Gauss. 

Let  x  represent  an  error  and  y  its  probability.     Then 


(1) 


(2) 


Let  us  suppose  that  n  observations  are  taken  on  the  quan- 
tities si,  and  22.  The  most  probable  values  of  these  quan- 
tities will  make  P  maximum, 


For  compound  probability 
P  =  2/12/2  .  .  .  2/» 


dP 


df(x2) 


dP    _  df(xi)  +_df(x2) 


df(xj 

'  /(*.)&! 

df(xa) 


(4) 


We  have 


.    .    .    .     (5) 


PROBABILITY  11 

Since  if  we  differentiate  log/(zi)  with  respect  to  xi,  we  have 


Substituting  (5)  in  (3)  and  (4),  we  have 

^7/*»  ..  x7/"h«  -.  y-7/V» 

r=0    •    •    (6) 


"2 


,  /        \^^n_rt  /ys 


The  equations  may  be  simplified  as  follows:  Consider 
that  z  has  been  measured  n  times,  giving  mi,  m^  .  .  .  mn 
as  results.  Then 

z\ — mi  =2/1 


z\—mn—xn 
Since  mi,  W2,  and  wn  are  constants,  we  have 

dxi     1      dx2     1      dxn     1  /0x 

^—  =  1;    -j— =1;    T~=l>    ....     (8) 
d^i  d^i  dzi 

and  similar  results  follow  for  dz2. 

This  greatly  simplifies  Eqs.  (6)  and  (7)  and  produces 

=0.      ...     (9) 


12  THEORY  OF  MEASUKEMENTS 

Errors  and  Residuals 

We  must  now  make  a  brief  digression  in  order  to  illus- 
trate the  difference  between  errors  and  residuals.  Take 
the  following  measurements,  of  equal  weight: 

mi  =430.6 
w2  =  429.9 


w4  =  430.8 

By  universal  custom  the  mean  of  these  results,  430.6, 
is  taken  as  the  best  attainable  value.  (Later  a  mathe- 
matical proof  of  this  will  be  given.) 

The  differences  between  the  measurements  and  the 
mean  are  called  residuals,  and  are  as  follows: 

vi=  0.0 
2,2=  -0.7 
2,3  =+0.5 


The  algebraic  sum  of  the  residuals  always  equals  zero, 
and  this  may  serve  as  a  check  upon  the  work.  Now  if  we 
had  some  way  of  knowing  that  430.6  was  the  correct  result, 
we  would  transform  those  residuals  into  errors  and  desig- 
nate them  by  xi,  X2,  £3,  and  x±. 

A  residual,  then,  is  the  difference  between  a  measure- 
ment and  the  best  attainable  result;  an  error  is  the  dif- 
ference between  a  measurement  and  the  true  result. 


PROBABILITY  13 

It  is  obvious,  first,  that  we  cannot  determine  the  values 
of  the  errors;  and  secondly,  that  the  sum  of  the  errors 
will  approach  the  sum  of  the  residuals  as  we  increase  the 
number  of  measurements. 

Going  back  to  Eq.  (9),  we  may  assume  that  n  is  suf- 
ficiently large  so  that  we  can  apply  the  above  law.  Then, 
since  the  sum  of  the  residuals  is  always  zero,  one  may 
take  the  sum  of  the  errors  to  be  zero,  and  write 


From  this  it  follows  that  <£  is  a  constant,  which  we  may 
call  c. 

If  we  substitute  the  values  of  <j>(xi),  $(#2),  and  (j>(xn) 
for  (5)  in  (9),  we  obtain 


-„  |  CT,  | 


Since  this  holds  for  any  number  of  observations,  the 
corresponding  terms  are  equal.  Omitting  subscripts  we 
have  in  general 


ex. 


f(x)dx 
Substitute  values  from  (1)  and  integrate.    We  have 


14  THEORY  OF  MEASUREMENTS 

c  is  negative  (why?)  and  may  be  replaced  for  the  sake  of 
convenience  by  —  2h2.  We  may  also  replace  ek'  by  k. 
Then  y  =  ke~h***,  which  is  the  equation  of  the  probability 
curve.  It  is  similar  in  form  to  the  equation  suggested  to 
express  the  curve  in  Fig.  1. 

This  is  the  most  important  equation  in  the  theory  of 
precision  of  measurements  and  its  meaning  should  be 
clearly  understood. 

PROBLEMS 

1.  Show  from  the  curve  that  positive  and  negative 
errors  are  equally  likely  to  occur. 

2.  Show  that  k  represents  the  probability  of  the  error 
zero. 

3.  Explain  why  h  is  called  the  measure  of  precision. 

4.  Show  that  the  curve  is  horizontal  over  the  origin. 

5.  Show  that  a  point  of  inflection  occurs  when 


THE  PROBABILITY  INTEGRAL 

In  order  to  express  the  probability  that  a  certain  group 
of  errors  will  be  made  we  integrate  between  the  limits 
concerned.  As  this  involves  compound  probability,  we 
have 


p  =  ^_|      e-tofifa. 

For  certainty  P  =  1  and  the  limits  become  +  oo  and  —  oo . 


PROBABILITY  15 

This  is  a  well-known  integral  and  is  discussed  in  various 

\A 

h 


treatises  on  the  calculus.     It  equals  -7—,  and  therefore 


_  1  _hdx 

L  —     7    7   •        A/  —      ,  —  . 
dxh  '  VTT 

and  the  probability  equation  becomes 


For  compound  probability 


The  probability  that  an  error  lies  between  +x  and  —x 
is  double  the  probability  that  it  lies  between  x  and  0.  We 
have 


This  may  take  the  form 


This  is  the  usual  form  of  the  probability  integral,  h  may 
be  determined  in  a  manner  to  be  explained  later;  then  by 
use  of  a  table  the  probability  of  any  desired  magnitude 
may  be  obtained.  The  student  is  referred  to  any  of  the 
larger  text-books  on  Least  Squares  for  illustrations. 


16  THEORY  OF  MEASUREMENTS 

EXERCISES 

In  order  to  become   familiar  with  the  terms  h  and  k, 
the  following  curves  should  be  plotted: 

1.  Consider  h  constant.     Give  k  the  values  1,  2,  3,  4. 
Plot  a  curve  for  each  value. 

2.  Consider  k  constant.     Give  h  the  values  },  \,  1,  2. 
In  each  case  select  a  series  of  rather  small  numbers 

for  x  and  use  the  formula 


3.  If  time  permits,  a  set  of  curves  should  be  con- 
structed in  which  both  h  and  k  are  variables,  x  and  y 
having  constant  values. 

The  Term  "  Least  Squares  " 

If  we  take  the  product  of  a  number  of  single  probabili- 
ties, we  obtain 


An  inspection  of  this  equation  shows  us  that  the  probabil- 
ity is  greatest  when  the  expression  in  the  parenthesis  is  least. 
But  these  numbers  are  the  squares  of  the  errors  (residuals, 
see  page  12).  It  follows  that  the  most  probable  values 
of  observed  quantities  are  those  which  make  the  sum  of 
the  squares  of  the  residuals  the  least.  From  this  law  is 
derived  the  term  Least  Squares. 


PEOBABILITY  17 

THE  ARITHMETICAL  MEAN 

We  are  now  prepared  to  prove  that  in  a  set  of  observa- 
tions, the  arithmetical  mean  has  the  greatest  probability. 

Let  M  be  the  mean  of  observations  mi,  m%  .  .  .  mn. 
Then 

M—mi,    M — m<2,  .  .  .  M — mn  are  residuals. 
We  may  make  the  sum  of  their  squares  a  minimum. 


.  .  .  2(M-mn)=Q. 

M  =  "vi  '  "**  '  "*n,  the  arithmetical  mean. 

n 

A  CONSTANT  INTERVAL.  FORMULA 


n(n2—  1) 

For  a  constant  interval  the  arithmetical  mean  should  not 
be  employed.  This  formula  comes  from  formulae  used  in 
developing  normal  equations.  See  Mellor's  Higher  Math- 
ematics, page  327. 

If  n  is  odd,  the  middle  term  does  not  appear.  Use  an 
even  number  of  measurements. 


18  THEORY  OF  MEASUREMENTS 

ILLUSTRATIVE    PROBLEMS 

1.  Kundt's   experiment:    30.7,   43.1,    55.6,    67.9,    80.1, 
92.3,    104.6,   116.9,   129.2,    141.7,    154.0,    166.1   cm.     Ans. 
12.3  cm. 

2.  Time  of  vibration  of  magnet  bar:    3.25,  9.90,  16.65, 
23.35,  30.00,  36.65,  43.30,  50.00,  56.70,  63.30,  69.80,  76.55, 
83.30,  89.90,  96.65,  103.15,  109.80,  116.65,  123.25,  129.95, 
136.70,  143.35.     Ans.  6.67. 

3.  A  problem  illustrating  this  method  may  easily  be 
suggested   from   measurements   with    a    planimeter;     also 
from  the  acceleration  curve  with  a  dropped  fork, 

WEIGHTS 

Heretofore  we  have  considered  all  our  measurements 
to  be  of  equal  value.  Suppose  the  length  of  a  small  object 
was  measured  with  a  micrometer  gauge,  a  vernier  reading 
to  hundredths  of  a  millimeter,  a  vernier  reading  to  tenths 
of  a  millimeter,  and  a  meter  stick. 

The  following  results  might  be  recorded: 

1.  Micrometer 3.542  cm. 

2.  Vernier  A 3.544   " 

3.  Vernier  B 3.54     " 

4.  Meter  Stick 3.55     " 

The  mean  of  these  measurements,  3.544,  is  obviously 
not  the  best  value,  since  some  represent  greater  precision 
than  others.  If  one  attaches  to  these  results  a  number 
indicating  their  relative  values,  such  a  number  is  called  the 


PROBABILITY 


19 


weight  and  is  usually  designated  by  p.  (Latin,  pondus, 
weight.  Compare  pound.)  Further  on,  rules  will  be 
given  for  finding  the  weight  of  observations,  but  for  the 
present  we  may  use  our  judgment.  No.  4  is  evidently 
the  least  accurate  and  we  may  give  it  the  weight  1.  No. 
3  comes  next  and  we  may  call  it  3.  No.  1  and  No.  2  are 
about  alike  in  weight,  with  perhaps  a  little  advantage  in 
favor  of  No.  1.  We  may,  therefore,  assign  8  to  No.  2  and 
10  to  No.  1.  Putting  our  results  in  tabular  form,  we  have: 


No. 

Obs.  (m) 

Weights  (p) 

Weighted  Obs.  (pm) 

1 

3.542 

10 

35.420 

2 

3.544 

8 

28.352 

3 

3.54 

3 

10.62 

4 

3.55 

1 

3.55 

=  77.942 
Spra  -T-  Sp = weighted  mean =3.543 

The  weighted  mean  may  be  proved  to  have  the  greatest 
probability  in  a  manner  similar  to  that  employed  for  the 
unweighted  mean. 

PROBLEMS 

1.  In  establishing  a  north  and  south  line,  the  following 
readings  were  taken: 

N.  6'  E.,  N.  4'  W.,  N.  50"  W.,  N.  00'. 

All  readings  were  taken  with  equal  care,  but  the  first  was 
taken  twice  and  the  third  three  times.  Find  the  best 
value. 


20  THEORY  OF  MEASUREMENTS 

2.  Joule's  values  of  the  mechanical  equivalent  of  heat 
have  been  weighted  by  Rowland  as  follows: 


442.8  (0); 
428.7  (2); 
425.8  (2); 
426.0  (5); 

427.5  (2); 
429.1  (1); 
428.0  (3); 
422.7  (1); 

426.8  (10); 
428.0  (1); 
427.1  (3); 
426.3  (1). 

He  concludes  that  426.9  best  represents  the  result  of  Joule's 
work. 

3.  The  sum  of  the  angles  of  an  equilateral  triangle  is 
found  to  measure  180°  9'.  If  the  first  angle  has  been  meas- 
ured six  times,  the  second  three  times,  and  the  third  once, 
how  should  the  error  be  distributed  among  the  angles? 

Additional  problems  in  weighting  will  be  given  after 
the  subject  of  precision  of  measurement  has  been  dis- 
cussed. 

Subjects  for  Discussion 

The  following  topics  are  suggested  for  those  who  may 
wish  to  pursue  the  subject  further: 

1.  The  part  played  by  the  theory  of  probability  in  the 
work  of  the  U.  S.  Weather  Bureau. 

2.  The  bearing  of  the  laws  of  probability  upon  popular 
superstitions.     Read  the  Vice-Presidential  address  of  Pro- 
fessor A.   G.   Webster,   at  Atlanta,   Ga.,  Dec.   30,  1913. 
(Science,  Jan.  9,  1914.) 

3.  Exceptional  phenomena.   Read  Chapter  29  in  Jevons's 
Principles  of  Science. 

4.  Relation  to  gambling. 


PROBABILITY  21 

Dr.  H.  G.  Burnham,  of  Chicago,  thinks  that  the  best 
way  to  wipe  out  gambling  in  America  is  to  teach  the 
children  in  the  schools  the  laws  of  chance.  He  feels  sure 
that  the  result  of  this  would  be  that  in  childhood  they 
would  steer  clear  of  the  slot  machine,  and  that  when  they 
grow  up,  they  would  shun  the  book  maker  and  every  other 
gambling  magnate. 

To  quote  further:  "  The  ordinary  gambler  does  not 
stop  to  count  his  chances  when  he  sees  that  by  betting 
a  dollar  he  may  win  one  hundred  dollars.  If  he  had  been 
taught  in  school  to  see  that  in  reality  the  chances  were 
200  to  one  against  him,  and  that  he  was  betting  a  dollar 
against  fifty  cents,  he  would  keep  his  money  in  his  pocket." 

5.  The  use  of  the  theory  of  probability  in  mortality 
tables. 


CHAPTER  III 
THE  ADJUSTMENT  OF  OBSERVATIONS 

IT  is  a  frequent  experience  in  making  measurements 
that  our  results  do  not  check.  The  measurements  may  be 
conditioned  (see  page  2).  The  sum  of  the  angles  of  a 
triangle,  for  example,  may  not  come  out  just  180°.  Or 
they  may  be  measurements  which  have  a  less  degree  of 
interdependence. 

If  we  take  0  as  the  starting-point  and  measure  a  dis- 
tance, mi  =  6.2  feet  above  0;  then  W2=4.3  above  mi; 
then  m2  =  10.6  above  0;  we  observe  that  there  is  a  discrep- 
ancy somewhere.  In  this  case,  an  obvious  method  of 
adjustment  would  be  to  add  0.03  to  the  first  and  second 
observations  and  subtract  it  from  the  third.  Since  the 
discrepancy  is  0.1,  this  adjustment  would  correct  it 
to  0.01. 

Theory.  While  this  method  would  answer  very  well 
for  the  simple  case  in  question,  it  would  break  down  when 
applied  to  more  extended  measurements.  The  theory 
of  the  method  of  adjustment  is  as  follows: 

Suppose  we  have  unknown  quantities  mi,  m^  .  .  .  mn, 
and  suppose  n  measurements  are  made  upon  these  quan- 

22 


THE  ADJUSTMENT  OF  OBSERVATIONS 

titles.     Then  if  a,  b,  c,  ...  Z  are  known  constants  and 
M  the  resulting  measured  quantity,  we  have 


Since  there  are  more  equations  than  there  are  unknowns 
it  follows  that  in  general  no  system  of  values  will  exactly 
satisfy  them.  Each  equation  has  a  most  probable  value 
for  its  unknown  terms,  but  in  each  case  there  will  be  left 
a  small  residual  v.  We  may  write  the  equation: 


For  simplicity  we  may  designate  all  the  terms  inde- 
pendent of  mi  by  K. 

=  vi,  and  by  symmetry 


Square  both  sides  of  these  equations  and  add: 


24  THEORY  OF  MEASUREMENTS 

The  greatest  probability  occurs  when  the  right-hand 
member  is  a  minimum.  Placing  the  first  derivative  equal 
to  zero,  we  have 

=0. 

Similar  equations  may  be  written  for  W2  .  .  .  m». 
These  are  called  "  normal  equations  "  and  their  solution 
gives  the  most  probable  values  of  the  observations  under 
consideration. 

Rules.  We  may  now  summarize  the  methods  of  adjust- 
ing observations  as  follows: 

1.  Write  an  observation  equation  for  each  observation. 

2.  Form  a  normal  equation  for  each  unknown  by  multi- 
plying each  observation  equation  by  the  coefficient  of  the 
unknown  in  that  equation  and  adding  the  results. 

3.  Solve  the  normal  equations  by  any  method.     These 
results  are  the  most  probable  values. 

i  After  these  "  best  values  "  have  been  found,  the  residuals 
should  be  computed,  and  the  sum  of  their  squares  (2y2) 
found.  This  value  is  less  than  that  for  any  other  possible 
values  of  the  residuals. 

ILLUSTRATIVE    PROBLEM 

1.  Given  the  following  observation  equations, 


=  —3 
mi  -f  2ra2  +  2ms  =  25 


m\-\-  7772+^3  =  16 


THE  ADJUSTMENT  OF  OBSERVATIONS  25 

Find  the  best  values  and  the  sum  of  the  squares  of  the 
residuals. 

We  first  find  the  normal  equations  by  the  rule.     For  mi, 


=  —  9 
25 
ra2+  m3=     16 


=     32 
For  W2, 

—  12wi  -f-  16w2  —  4ms  =  12 
2wi-f- 


For 


=—  3 
=  50 
=  10 
=  16 


6mi+2m2+7m3=     73 
Grouping  the  normal  equations, 


26  THEORY  OF  MEASUREMENTS 

Solving, 

mi  =  6.00 
m2  =  6.13 
m3  =  3.26 

To  obtain  the  residuals,  we  substitute  these  values  in 
the  observation  equations, 

18-24.52+3.26=-  3  t>i  =     0.26 

6+12.26+6.52=     25  t;2=-0.22 

6.13+  3.26  =10  t>3=     0.61 

6+  6.13+3.26=     16  v4=     0.61 

2v2  =  0.860 

It  would  be  an  interesting  exercise  to  try  other  values 
for  mi,  W2,  and  mz  and  compare  the  value  of  Zy2  resulting, 
with  the  above  values.  If,  for  example,  we  choose  mi  =  6, 
W2  =  6,  ma  =  3,  it  is  obvious  that  the  second  residual  is  unity, 
which  alone  is  greater  than  the  value  of  2v2. 

The  following  observations  were  made  on  a  triangle: 

Angle  A  =45°;  angle  £  =  80°;  angle  C  =  54°.  A+B 
=  126°;  A+<7  =  100°;  A+£+C  =  180°.  Find  best  results. 

The  observation  equations  are 

mi  =  45 


The   student   should   construct   the   normal   equations 
and  find  the  best  values  for  mi,  m2,  and  m^. 


THE  ADJUSTMENT  OF  OBSERVATIONS  27 

MEASUREMENTS  OF  A  LINE 

B  C  ~     i~~          F  ~ 

FIG.  2. 


=4.Q  units  FG=  7.8  units 

BC  =  5.l  AC=  9.0 

CD  =  6.9  BD  =  11.9 

DE  =  2.0  DF=  8.2 

#^  =  6.1  A£=32.0 

Find  the  best  values  of  the  above  measurements  made 
along  a  line. 

Here  we  have  10  observation  equations  which  will 
reduce  to  six  normal  equations  containing  six  unknown 
quantities,  which  may  be  solved  in  the  usual  way.  The 
problem  is  inserted  for  the  purpose  of  illustration;  its 
solution  is  hardly  worth  while. 

FORMULAE 

When  the  number  of  equations  is  large  and  the  con- 
stants are  not  whole  numbers,  it  saves  time  and  affords  a 
check  upon  the  work  to  use  the  following  formulae: 

.  .  an2 


28  THEORY  OF  MEASUREMENTS 

The  normal  equations  become 


bami+bbm2+  .  .  . 


almi-\-blm2-\-  .  .  .  Urn*  =  IM 

The  following  example  will  illustrate  the  use  of  these 
formulae  : 

Observation  Equations 


—  3w2  =  15.1 
2wi+2ra2  =  13.9 
mi—  2ra2  =  2.1 

aa=  +36+4+1  =  +41 

a&=-18+4-2=-16 

66  =  +9+4+4=  +17 
aM=  +90.6+27.8+2.1  =  +120.5 
bM=  -45.3+27.8  -4.2  =-21.7 

41mi+16w2  =  120.5 
16mi+17m2=-21.7 

Let  the  student  compare  these  results  with  those  found 
without  the  use  of  the  formula. 


THE  ADJUSTMENT  OF  OBSERVATIONS 


29 


PROBLEMS 

1.  Adjust  the  following  values  and  find 

mi  =  +3.06 
m2=  -1.30 


2mi+ra2=+4.81 


2.  The  following  observations  are  taken  from  a  student's 


field  book: 


M  =  147°  14'  04.5" 

N=  88    58  06.1 

0=  93    51  26.7 

P=  29    56  16.6 

59  55.0 

3  39.2 

45  53.0 

1  57.8 

8  30.0 


FIG.  3. 
Find  the  best  values  of  the  angles. 


30 


THEORY  OF  MEASUREMENTS 


A  =   22°  13'  59" 

5  =  100  18  40 

C=  57  27  33 

=  122  32  41 

£+(7  =  157  46  10 

C+A=  79  41  29 

180  00  3 


FIG.  4. 

Find  the  best  values  of  the  angles. 
4.  Adjust  the  following  angles: 


BOC  =  2 
COD  =  3 
DOE  =  4 


FIG.  5. 


THE  ADJUSTMENT  OF  OBSERVATIONS  31 

No.  Observed  Angles. 

1 70°  20'  51" 

2 52  35  10 

3 '. 50  41  25 

4 95  10  41 

5 v 91  11  57 

6 .,....; 198  27  14 

Observe  that  angle  6  =  angle  2-f-angle  3+angle  4. 
5.  Clairaut's  empirical  formula  for  the  relation  between 
the  length  of  a  seconds  pendulum  and  the  latitude  is 

l=L0+A  sin2L. 

Suppose  the  following  observations  to  be  made: 

L  i 

0°    0'  0.990564 

18   27  0.991150 

48    24  0.993867 

58    15  0.994589 

67     4  0.995325 

Substitute  the  values  in  the  given  equation  for  observa- 
tion equations;  then  form  normal  equations  for  LQ  and  A. 
Mellor  gives  these  normal  equations: 

0.993099 =L0+0.44765A, 
0.994548  =  L0-|-0.70306A. 


32  THEORY  OF  MEASUREMENTS 

SHORTER  METHODS 

The  following  abbreviated  methods  are  sometimes 
employed  : 

MAYER'S  METHOD 

Make  all  the  coefficients  of  mi  positive  and  add  the 
results  to  form  a  normal  equation  for  mi.  Similarly  for 
m,2  .  .  .  mn.  Solve  the  normal  equations  as  before. 

The  results  are  not  so  accurate  as  those  obtained  by  the 
longer  method,  but  are  satisfactory  for  most  purposes. 

PROBLEMS 

Observation  equations. 

x-y+2z=  3 
-5z=  5 


Solve  by  the  regular  method  and  by  Mayer's  method. 
Some  idea  of  their  relative  accuracies  may  be  obtained 
by  computing  Zv2  in  each  case. 

METHOD  BY  DIMINISHING  THE  CONSTANT  TERM 
Suppose  we  have  given 

A+B=W 

4A-B  =  19 

2A  +35  =  25 

An  inspection  of  these  equations  shows  that  A  =  6  and 
B  =  4  approximately. 


THE  ADJUSTMENT  OF  OBSERVATIONS  33 

Let  a  and  b  equal  the  difference  between  the  true  and 
approximate  values  of  A  and  B.  Then 

(6+a)  + (4+6) -10  =  0 

4(6+a)-  (4+6)  -19  =  0 

2(6+a)  +3(4+6)  -25  =0 

a+6  =  0 

4a-6+l  =  0 

2a+36-l=0 

Forming  normals  and  solving,  we  have 

o=-0.153 
b=  +0.406 
A  =6-0. 153  =  5.847 
5  =  4+0.406=4.406 

The  student  should  compare  these  results  with  those 
obtained  by  the  regular  method. 

This  method  is  especially  well  adapted  to  the  adjust- 
ment of  angles  observed  at  a  station.  Considerable 
mathematical  computation  is  avoided.  The  following 
problem  from  the  United  States  Lake  Survey  is  solved 
in  Merriman's  Least  Squares. 

No.  Between  Stations.  Observation. 

1.  Bunday  and  Wheatland 44°  25'  40//.613 

2.  Bunday  and  Pittsford 80  47  32  .819 

3.  Wheatland  and  Pittsford '. .  36  21  51  .996 

4.  Pittsford  and  Reading 91  34  24  .758 

5.  Pittsford  and  Bunday 279  12  27  .619 

6.  Reading  and  Quincy 62  37  43  .405 

7.  Quincy  and  Bunday 125  00  18  .808 


34  THEORY  OF  MEASUREMENTS 


We  may  set  up  the  observation  equations: 


mi  =  44°  25'  40".613 

mi+m3=  80  47  32  .819 

w3  =  36  21  51  .996 

7?i4=  91  34  24  .758 

360°-(wi+m3)=279  12  27  .619 

w6=  62  37  43  .405 

125  00  18  .808 


FIG.  6. 


On  account  of  the  nature  of  the  constant  terms  the 
solution  would  be  extremely  tedious.  We  may  let  vi, 
vs,  v±,  and  VQ  be  the  most  probable  corrections  to  be  applied. 
Then 

mi=44°25'40".613-{-t;i 
w3  =  36  21  51  .996 +v3 
m4  =  91  34  24  .758+^4 
m6  =  62  37  43  .405+^6 


THE  ADJUSTMENT  OF  OBSERVATIONS  35 

This  gives  rise  to  simpler  observation  equations: 

vi=Q 
^4-^3=4-0.210 


=-  0.228 


+0.420 

The  right-hand  members  denote  seconds. 
The  normal  equations  are 

=  +0.402 
=  +0.402 
=  +0.420 
=  +0.420 

From  which 


v3=+0  .022 
t;4=+0  .126 
v6  =  +0  426 


The  adjusted  values  now  become, 

No.l  ....................     44°  25'  40r/.635 

2  ...........  ____  ...     80  47  32  .653 

3  ....................     36  21  52  .018 

4,  .........7.  ........     91  34  24  .884 

5  ..........  ..  .  ........  279  12  27  .347 

6,.  ............  .......     62  37  43  .531 

7..                                 ,  125  00  18  .932 


CHAPTER  IV 
THE  PRECISION  OF  OBSERVATIONS 

As  stated  in  the  introductory  chapter,  there  are  no 
measurements  made  with  such  a  degree  of  accuracy  that 
we  may  regard  them  as  absolutely  correct.  The  wave- 
length of  cadmium  light  has  been  measured  with  an  accu- 
racy which  is  marvelous,  but  its  exact  value  will  probably 
never  be  known.  Since  measurements  differ  among  them- 
selves in  accuracy,  it  is  desirable  to  have  some  method 
of  indicating  this  fact,  and  the  term  "  Precision  of  Obser- 
vations "  or  "  Precision  of  Measurements  "  is  applied  to 
the  method  of  procedure  employed  in  determining  the 
relative  accuracy  of  measurements.  Strictly  speaking, 
the  term  "  deviation  'Ms  a  better  one  than  "  accuracy  " 
because  the  latter  term  seems  to  presuppose  a  knowledge 
of  exact  results. 

Classes.    There  are  three  classes  of  precision  measures: 

1.  Mean  error; 

2.  Average  deviation; 

3.  Probable  error. 

Graphic  Method.  These  may  be  illustrated  graph- 
ically by  use  of  the  probability  curve  (Fig.  7). 

Since  YOX  may  be  regarded  as  a  probability  area, 
and  since,  as  will  be  shown  later,  the  probability  that 
the  true  result  lies  within  the  limits  indicated  by  the 

36 


THE  PRECISION  OF  OBSERVATIONS 


37 


precision  measure  is  one-half,  it  follows  that  we  may  express 
our  precision  in  one  of  three  ways: 

1.  By  taking  the  distance  from  0  along  the  z-axis  to 
the  point  of  inflection  of  the  curve.     This  is  represented 
by  OM  and  expresses  the  mean  error  or  mean  square  error. 

2.  By  taking  the  abscissa  of  the  ordinate  passing  through 
the  center  of  gravity.    This  is  OD  and  expresses  the  aver- 
age deviation. 


3.  By  taking  the  abscissa  of  the  ordinate  which  divides 
YOX  into  equal  parts.  This  is  OP  and  expresses  the 
probable  error. 

By  reference  to  Problem  5,  page  14,  it  will  be  seen  that 

the  mean  error,  /z=  -  —,  the  distance  OM  in  the  figure. 


The  average  deviation,  OD,  may  be  found  by  use  of  the 
formula  in  mechanics 


l/2h2 


38  THEORY  OF  MEASUREMENTS 

0  4769 
The  probable  error,  OP  —  ^—r — .     This  will  be  deduced 

later. 

Of  these  precision  measures,  it  may  be  said  that  the 
mean  error,  "  the  square  root  of  the  arithmetical  mean  of 
the  squares  of  the  errors,"  is  seldom  used;  the  probable 
error  is  the  most  accurate;  and  the  average  deviation  is 
the  easiest  to  determine. 

THE  AVERAGE  DEVIATION 

If  m  is  the  mean  of  n  measurements  and  v\,  V2,  #3,  .  .  . 
t>0  the  residuals  (neglecting  signs),  the  average  deviation 

would  be  a.d.  =  — . 
n 

Taking  the  numbers  10,  8,  9,  10,  the  mean  is  9.25, 
and  vi  =  0.75,  v2  =  1.25,  ^  =  0.25,  and  v4  =  0.75.  Sv  =  3.00 
and  a.d.  =  0.75.  We  may  neglect  the  signs  of  the  residuals. 

This  is  the  average  deviation  of  a  single  observation. 

The  average  deviation  of  the  mean,  A.D.  =  —7=.     In 

Vn 

the  problem  just  solved,  A.D.  =  — ^— =  0.38. 

a 

PROBLEMS 

It  will  prove  of  interest  if  the  instructor  will  assign 
problems  taken  from  the  student's  laboratory  note-book 
in  illustration  of  this  topic. 

THE  PROBABLE  ERROR 

The  term  probable  error  is  rather  misleading,  but  its 
meaning  may  be  made  clear  by  the  following  definition: 


THE  PRECISION  OF  OBSERVATIONS  39 

The  probable  error  is  a  number  placed  after  a  result  with 
a  plus  and  minus  sign  between  them;  and  it  indicates 
that  it  is  an  even  wager  that  the  true  result  lies  between 
the  indicated  limits,  and  that  it  does  not  so  lie. 

At  first  thought,  this  definition  seems  vague  and  not 
indicative  of  a  very  high  order  of  precision.  Let  us  take 
the  value  of  the  velocity  of  light  as  299860zb30  km.  per 
second.  This  means  that  it  is  just  as  likely  that  the  true 
value  lies  between  299830  and  299890  as  that  it  lies  between 
zero  and  299830,  plus  299890  to  infinity.  This  narrows 
the  field  and  indicates  an  order  of  precision  that  is  suf- 
ficiently high. 

FORMULAE  FOR  PROBABLE  ERROR 

Going  back  to  the  probability  integral,  we  must  take 
the  value  of  x  when  P  =  |.  This  gives  J  =  2/  vV  \  Xe~h2xSdhx. 

From  integration  tables  we  find  hx  =  0.4769. 
The  particular  value  of  x  which  fulfills  this  condition 
will  be  denoted  by  r.    Then  hr  =  0.4769. 

Take  the  equation  for  compound  probability  (page  15). 


Since  h  is  the  measure  of  precision,  we  wish  to  give  it 
such  a  value  as  will  render  P  a  maximum. 


ah 


40  THEORY  OF  MEASUREMENTS 

(Divide  by  ftn-i6- 


But  h=<^. 


Therefore       r  =  0.4769  J—  =  0.6745*  /—  . 

\    n  \  n 

This  is  the  probable  error  for  a  single  observation  when 
errors  are  considered. 

In  order  to  change  errors  to  residuals  (a;  to  v),  we  make 
use  of  the  following  procedure:  Let  the  sum  of  the  squares 
of  the  errors  differ  from  the  sum  of  the  squares  of  the 
residuals  by  some  constant,  say  u2.  Then 


Now  u2  =  —  : 

n 


n      n—1' 
Substituting  in  the  formula  for  r, 


=  0.6745- 

n— 


THE  PRECISION  OF  OBSERVATIONS  41 

This  is  the  formula  for  single  observations  for  residuals. 

Since  probable  errors  are  inversely  proportional  to  the 
squares  of  the  number  of  observations  taken  (see  any  text- 
book on  Least  Squares),  we  have 

n:  l::l/ro2:  1/r2, 
ro  =  r/Vn. 

ro  thus  represents  the  probable  error  of  the  mean  and 

/     Zy2 

may  be  written  ro  =  0.6745* /^ r-r. 

\n(n—  1) 

We  may  now  summarize  the  relation  between  the 
probable  error,  the  measure  of  precision,  and  the  weight  as 
follows: 

1.  The  measure  of  precision  varies  inversely  as  the 
probable  error  (since  hr  =  constant) . 

2.  Weights  are  proportional  to  the  squares  of  the  pre- 
cision measures. 

3.  Weights  are  inversely  proportional  to  the  squares 
of  the  probable  errors. 

PROBLEMS 

1.  Given  the  following  measurements  of  the  length  of 
a  line: 

70.6  cm.  70.5  cm.  70.4  cm. 
70.5    "                  70.6    "  70.5    " 

70.7  "  70.8    "  70.6   " 

Find  the  mean  square  error,  the  average  deviation, 
and  the  probable  error. 


42  THEORY  OF  MEASUREMENTS 

2.  Assume  that  the  lines  OP,  OD,  and  OM  are  drawn 
to  scale  in  Fig.  7,  and  compare  their  relative  lengths  with 
the  results  of  No.  1. 

3.  Given  the  measurements  and  probable  errors: 

427.32±0.04 
427.30±0.16. 

Find  the  relative  weight  and  the  relative  precision. 

4.  Twenty   measurements   of   a   line   give   a   probable 
error  of  the  mean  of  0.06.     How  many  additional  measure- 
ments are  required  to  reduce  the  probable  error  to  0.03? 

5.  An  angle  is  measured  9  times  with  each  of  two  tran- 
sits.   The  first  gives  a  value  of 

41°  32'  14"±8".2. 
The  second  gives 

41°  32'  12"±7".l. 

Find  the  best  value  for  the  angle. 

6.  When  the  best  value  is  found  from  No.  5,  its  prob- 
able error  may  be  found  by  use  of  the  formula 

r/2  :  ra2  ::  pa  :  pf 

where  r/=  probable  error  in  the  best  value; 
ra= probable  error  in  the  first  value; 
pa  =  weight  of  first  value; 
pf= weight  of  first  value + weight  of  second  value. 

The   precision   measure   is   sometimes   expressed   as   a 
fraction  or  a  decimal  with  reference  to  the  magnitude  of 


THE  PRECISION  OF  OBSERVATIONS  43 

the  measurement  concerned.  Thus  in  Problem  3,  we  may 
say  that  0.04  is  the  precision  measure,  or  we  may  say  that 
the  measurement  is  reliable  to  about  0.01%,  or  to  one  part 
in  10,000. 

PROBLEMS 

1.  Show  from  the  data  on  page  37  that  r=0.85  a.d. 
=  0.67M. 

It  follows  that  for  a  constant  value  of  n,  ro  =  0.85  A  .D. 

2.  The  following  results  have  been  obtained  from  nine 
measurements  in  each  case: 

94.31cm.  A. D.=  0.031 
9436  "  a.d.=  0.090 
94.35  "  r  =0.025 

94.33  "         r0=  0.007 

94.34  "    reliable  to  0.03% 

94.35  "    precise  to  2  parts  in  10,000 

Reduce   these   different    precision   measures    to  A.D.'s 
and  write  them  in  order  of  the  reliability  of  the  results. 


CHAPTER  V 
THE  PROPAGATION  OF  ERRORS 

IN  the  study  of  the  propagation  of  errors  we  have  two 
classes  of  problems  —  the  direct  and  the  converse.  In  the 
first  class  we  determine  how  the  errors  in  component 
measurements  affect  the  reliability  of  the  results;  and  in 
the  converse  problem  we  determine  with  what  accuracy 
we  should  make  our  component  measurements  in  order 
to  secure  a  required  accuracy  in  the  result. 

In  solving  problems  under  the  first  case,  we  proceed 
as  follows:  If  Z  represents  the  true  result,  and  X  its 
error;  and  z\,  22,  •  •  •  z»  are  true  values  of  component 
measurements,  and  xit  X2,  •  •  •  xn  are  their  corresponding 
errors,  then, 


,     (22+0:2)  .  .  • 
By  Taylor's  theorem 

v_dZ       dZ  dZ 

*  •  " 


Consider  that  we  have  a  series  of  such  terms,  designated 
by  SX,  Sa?i,  2^2,  etc.,  and  square  both  sides: 


44 


THE  PROPAGATION  OF  ERRORS  45 

We  have  neglected  terms  containing  the  products  of 
the  errors,  such  as  2x  1X2,  since  positive  and  negative  errors 
are  equally  likely  to  occur.  If  we  divide  both  sides  by  n, 
we  may  substitute  r2  (the  probable  error  in  the  final 


result)  for  -  and  ri2,  f22,  rn2,  the  probable  errors  of  'the 

Ti 


components  for  —  —  ,  etc.     The  reason  for  this  is  evident 

from  an  inspection  of  the  probable  error  formula.    Thus 
we  have 


Hereafter  we  shall  replace  r  by  A  and  r\,  r%  .  .  .  rn 
by  Si,  62,  ...  Sn.  These  refer  to  any  precision  measure 
as  well  as  the  probable  error.  A  means  the  deviation  in 
the  result  due  to  deviations  in  the  components,  Si,  fe, 


s«. 


ILLUSTRATIVE    PROBLEMS 


1.  The  two  sides  of  a  rectangle  measure  45±0.6  and 
50±0.4.    Find  the  deviation  in  the  area.    The  formula 


s 


=  (50)2(0.6)2+(45)2(0.4)2 
=  900+324  =  1224 
A  =  35  approximately. 

We  may  say,  then,  that  the  area   (expressed  in  square 
feet),  is  2250  with  a  deviation  measure  of  35. 


46  THEORY  OF  MEASUREMENTS 

2.  It  may  be  readily  seen  from  the  formula  that  if 
the  equation  has  the  form  A  =a+6,  the  value  of  A2  becomes 
5fl2-N62.     This    may  be  applied  to  the  measurement  of 
a  line  which  has  to  be  made  in  sections. 

3.  If    the    equation    contains    a    constant    multiplier, 
A=ca,  we  have  A  =  c5. 

If  the  radius  of  a  circle  is  10  cm.  with  a  deviation  of 
0.1,  the  deviation  of  the  resulting  length  of  the  circumfer- 
ence is  27rXO.  1  =  0.63. 

PROBLEMS 

1.  The  formula  for  the  radius  of  a  sphere  as  given  by 
a  spherometer  is 


If  Z  =  7.23±0.04cm., 

and  a  =  0.53±0.008  cm. 

find  the  length  of  the  radius  and  its  deviation  measure. 

2.  If  the  length  of  a  pendulum  is  lOOitO.l  cm.,  and  the 
period  of  one  vibration  1.01  ±0.003  seconds,  what  is  the 
deviation  in  0? 

3.  The  mass  of  a  body  in  air  is  30±0.1  g;    in  water 
20±0.2  g.    Find  the  deviation  in  the  value  of  its  specific 
gravity. 

4.  What  is  the  deviation  in  the  value  of  the  velocity 
of  liquid  flow  from  a  height  of  50dbl  cm.?     V2  =  2gh. 

,5.  A  mass  of  100  g.  is  revolved  with  a  speed  of  80  cm. 
per  second  on  the  end  of  a  corti  50  cm.  long.  The  mass 
measurements  are  101,  100,  99;  the  speed  measurements 


THE  PROPAGATION  OF  ERRORS  47 

82,  80,   78;    the  length  measurements  51,  49,  50.     Find 
the  centrifugal  force  and  its  deviation. 

6.  Find  the  deviation  in  the  mean  of  two  quantities 
which  differ  by  C. 

7.  If  d  is  the  deviation  in  log  a,  what  is  the  deviation 
in  a? 

8.  In  No.  2,  assume  that  the  deviation  found  is  an 
average  deviation,  A.D.    Express  the  result  fractionally, 
decimally,  and  as  a  probable  error. 

9.  In  determining  a  refractive   index,   the  value  of  i 
is  40°±8/,  and  of  r  32°±6/.     Find  the  value  of  the  index 
and  its  deviation. 

In  solving  these  problems,  the  student  should  observe 
the  relative  effects  of  the  deviations  in  the  components 
upon  the  result.  In  No.  2,  for  example,  it  will  be  seen 
that  a  deviation  in  t  is  a  much  more  serious  matter  than 
an  equal  deviation  in  I.  A  large  number  of  problems 
illustrating  this  process  may  be  found  in  Goodwin's  Pre- 
cision of  Measurements, 

PROBLEMS 

The  following  problems  are  of  especial  interest  to 
engineering  students : 

1.  In  the  triangle  ABC,  AB  =  500'd=0.04'  Z.A  =32°±10', 
ZC  =  60°±4'.  Find  BC  and  its  deviation.  The  formula 
for  the  area  of  a  triangle  with  the  given  conditions  should 
be  written  down  and  the  usual  method  employed.  We 
have  the  deviations  given  for  the  angles  and  need  them  for 
the  sines  of  the  angles.  For  a  deviation  of  10  minutes 
in  32  degrees,  we  find  the  deviation  in  the  sine  of  32  degrees 


48  THEORY  OF  MEASUREMENTS 

by  subtracting  the  sine  of  32°  from  that  of  32°  10'.  Or 
we  may  differentiate  with  respect  to  the  angle. 

2.  In    a    triangle    ABC,    we    have    A B  =  240.4 ±0.04, 
AC  =  290.6± 0.003,    Z.4=44°  20'±1'.     Find  the  area  and 
its  deviation.     Use  both  methods  referred  to  above  in  deal- 
ing with  the  angle  A. 

3.  The  formula  for  the  elastic  modulus  of  a  rectangular 

PI3 
bar  supported  at  its  ends  is  E=        3>  where  P  is  the  mass 

applied  at  the  center,  I  the  length,  d  the  deflection  produced, 
b  and  h  the  breadth  and  thickness.  We  have  given  the 
following  measurements : 

b  h                            d 

0".331  0".490  0".206 

0".333  0".492  0".205 

0".329  0".491  0".206 

0".330  0".490  0".207 

Take  the  length  two  feet  and  P  =  40  Ibs. 
Find  E  and  its  deviation. 

4.  Find  the  number  of  calories  and  its  deviation,  from 
a  current  of  6.2±0.06  amperes,  through   a  resistance  of 
20±0.1  ohms,  for  30  minutes  (correct  to  a  single  second). 

5.  Observations  on  a  resistance  of  about  10  ohms  give 
values  as  follows:    Correct  to  0.09%,  correct  to  one  part 
in  one  hundred;    a   probable  error  of  0.001.     Write  these 
down  in  the  order  of  their  relative  reliabilities. 


THE  PROPAGATION  OF  ERRORS  49 

THE  CONVERSE  PROBLEM 

In  the  cases  already  considered,  we  have  discussed  the 
effects  of  deviations  in  component  measurements  upon  the 
deviation  in  the  result.  We  are  commonly  called  upon  to 
determine  in  advance  to  what  degree  of  precision  we  shall 
make  our  measurements  in  order  to  reach  a  given  precision 
in  our  result.  For  example,  we  are  required  to  measure 
the  volume  of  a  cylinder  whose  length  is  about  10  cm.  and 
whose  radius  is  about  3  cm.  with  such  a  degree  of  accuracy 
that  the  deviation  in  the  volume  (about  283  cm.3)  shall 
not  exceed  3  cm.3  It  is  customary  to  so  adjust  the  devia- 
tions pertaining  to  each  variable  that  each  will  have  an  equal 
effect  upon  the  final  result.  This  is  called  the  method 
of  equal  effects.  Representing  the  effects  of  the  various 
deviations  upon  the  results  by  Aa,  A&,_.  .  .  An,  we  have 
A2=Aa2+A&2+  .  .  .  An2  =  nA02.  A  =  AaVr&.  In  our  problem 
we  may  write 

r~A/^~V2~ 

The  deviations  in  the  length  and  radius  should  not 
separately  produce  a  deviation  in  the  area  of  more  than 
2.1  cm.3  From  our  general  equation,  page  45,  we  have 


Therefore  dl  =  0.074  cm.     Likewise 


A,  =  ^r:5r  =  27rrZSr=  188.63,  =  2.1.     dr  =  0.011  cm. 
dr 


50  THEORY  OF  MEASUREMENTS 

We  conclude  that  we  must  measure  the  length  of  the 
cylinder  with  an  accuracy  of  0.074  cm.,  and  the  radius 
with  an  accuracy  of  0.011  cm. 

THE  FRACTIONAL  METHOD 

These  problems  may  be  solved  in  another  way.  It 
follows  (not  quite  directly)  from  the  method  of  equal 
effects  that  the  deviation  in  a  final  result  due  to  a  devia- 
tion in  a  component  bears  the  same  relation  to  the  final 
result  as  the  deviation  in  the  component  bears  to  the  com- 
ponent, if  the  exponent  is  unity. 

In  other  words  ~r—~r-    If  ^ne  component    has   any 

other  exponent  than  unity,  the  fraction  must  be  multiplied 
by  the  exponent,  following  the  method  of  differentiation. 

We  have  in  the  last  problem  -j-  =  —  -,    since    the   radius 

A.          T 

appears  in  the  second  power. 

A 
Since  Aj  =  Ar  =  —  ;=  we  may  write 

v2 

Ai    Ar       1    A       1      3 
Z  =  3 

=  0.0074;    dl=  0.074 


. 
This  is  seen  to  check  with  the  former  method. 


THE  PROPAGATION  OF  ERRORS  51 

PROBLEMS 

1.  With  what  accuracy  must  we  measure  the  radius 
of  a  circle  to  obtain  an  accuracy  of  0.1%  in  the  area? 
(First  method.) 

2.  With  what  accuracy  should  I  and  t  be  measured  in 
a  simple  pendulum  to  obtain  a  value  of  g  correct  to  one 
unit?     (Second  method.) 

3.  With  what  accuracy  should  h  and  a  be  measured 
with  a  spherometer  to  give  a  value  in  the  radius  correct 
to  0.03%?     This  problem  may  be  solved  in  a  literal  form 
or   values  may  be  assigned    to  a  and  6  and  the  result 
computed.     (First  or  second  method.) 

These  are  illustrative  problems.  They  should  be 
extended,  so  far  as  time  permits,  to  include  various  problems 
from  the  student's  laboratory  note-book.  The  impor- 
tance of  this  part  of  the  subject  is  obvious. 


BEST  MAGNITUDES  AND  BEST  RATIOS 

No  one  can  work  very  long  in  a  physical  laboratory 
without  observing  that  the  results  come  out  much  better 
by  using  certain  magnitudes  and  ratios  than  they  do  with 
others.  In  measuring  current  with  a  tangent  galvanom- 
eter very  poor  results  would  be  obtained  for  angles  between 
0°-20°  and  70°-90°.  Around  45°,  however,  they  are 
satisfactory.  In  using  a  slide-wire  Wheatstone  bridge, 
the  two  segments  of  the  wire  must  be  about  equal  for 
satisfactory  results. 


52  THEORY  OF  MEASUREMENTS 

In  the  first  case,  our  formula  is  I  =  c  tan  <j>. 

_  d(c  tan  <f>)  cd<t> 

dj          *     cos2  <£' 

Dividing 


/      cos2  <f>    c  tan  </>     sin  2<£* 

This  is  a  minimum  when  <£  =  45°  and  shows  that  the  devia- 
tion in  the  result  due  to  a  deviation  in  $  is  least  at  45°. 

For  the  Wheatstone  bridge  we  may  let  x  =  the  unknown 
resistance,  R  the  known,  a  and  b  the  segments  of  the  wire, 
and  c  its  total  length. 

Then 

aR 


x  = 


c—a 


It  should  be  easy  for  the  student  to  prove  that  the  best 
result  occurs  when  a  =  b. 

These  illustrations  might  be  indefinitely  extended. 
The  student  is  referred  to  Holman's  Precision  of  Measure- 
ments for  a  number  of  interesting  problems. 

EXERCISES 

1.  Show  that  the  relative  error  in  measuring  the  area 
of  a  circle  decreases  with  increasing  radius. 

2.  In  drawing  a  simple  harmonic  motion  curve,  discuss 
the  effect  of  an  error  in  the  phase  upon  the  displacement 
for  various  points  on  the  curve. 


THE  PROPAGATION  OF  ERRORS  53 

3.  A  body  rolls  down  an  inclined  plane  for  a  definite 
period  of  time.     Discuss  the  effect  of  an  error  in  measur- 
ing the  angle  of  the  plane  upon  the  error  in  velocity. 
Neglect  friction. 

4.  How  would  an  error  of  one  minute  in  measuring  the 
angle  of  incidence  compare  with  a  similar  error  in  measur- 
ing the  angle  of  refraction  in  their  effect  upon  the  refrac- 
tive index?    Take  z=45°  and  r  =  30°. 

5.  What  is  the  "  best  value  "  of  the  limiting  angle  in 
measuring  the  coefficient  of  friction? 

ABC 


6.  In  Lami's  theorem 


sin  a     sin  0     sin  7' 


take         A  =59.8,   £  =  69.8,    C=  102.8, 
a  =  146°,    0=139°,    7  =  75°. 

Discuss  the  effect  upon  each  term  of  an  error  of  0°.5  in 
measuring  a,  /3,  and  7. 


CHAPTER  VI 
PLOTTING 

Definitions.  It  may  be  assumed  that  students  for 
whom  this  book  was  prepared  have  a  knowledge  of  the 
fundamental  principles  of  curve  tracing.  A  few  familiar 
definitions  and  illustrations  will,  however,  be  given. 

A  plot  is  a  graphic  representation  of  the  relation  of 
two  quantities  of  which  one  is  a  function  of  the  other. 
The  origin  is  the  point  of  departure  from  which  all  distances 
are  reckoned.  The  coordinates  are  horizontal  and  vertical 
distances  from  the  origin.  The  axis  of  abscissas  is  the 
horizontal  line  through  the  origin.  The  axis  of  ordinates 
is  the  vertical  line  through  the  origin.  The  slope  of  the 
curve  is  the  angle  it  makes  with  the  z-axis. 

When  the  points  for  a  curve  have  been  established, 
there  are  two  methods  of  procedure:  If  the  curve  appears 
to  be  regular  or  to  correspond  with  well-known  types, 
we  should  draw  an  average  line  through  the  points  in  such 
a  manner  that  about  as  many  will  lie  on  one  side  as  on 
the  other.  If  the  curve  does  not  fulfill  these  conditions, 
the  adjacent  points  should  be  connected  by  straight 
lines. 

54 


PLOTTING 


55 


EXERCISES 

In  these  exercises,  the  student  should  use  his  judgment 
as  to  the  method  to  be  followed. 

1.  Plot  a  simple  interest  curve  for  $100  for  ten  years 
at  5  per  cent. 

2.  Plot  a  compound  interest  curve  starting  from  the 
same  origin. 

3.  The  maximum  temperature  for  each  day  of  a  certain 
month  was  as  follows: 


1 

31° 

11 

29° 

21 

11° 

2 

22 

12 

24 

22 

13 

3 

21 

13 

18 

23 

18 

4 

27 

14 

12 

24 

41 

5 

29 

15 

14 

25 

43 

6 

26 

16 

26 

26 

38 

7 

22 

17 

28 

27 

30 

8 

28 

18 

28 

28 

36 

9 

29 

19 

19 

29 

36 

10 

28 

20 

16 

30 

47 

Plot  the  curve. 

4.  Compute  the  mean  temperature  for  the  month  in 
No.  3,  and  plot  a  curve  showing  the  departure  from  the  mean 
for  each  day.  ^.  ; 

5.  The  following  plot  will  afford  amusement  as  well 
as  instruction: 


56 


THEORY  OF  MEASUREMENTS 


X 

V 

X 

V 

-0.0 

1.5 

1.3 

-1.0 

-1.0 

1.2 

1.6 

-0.3 

-2.0 

0.7 

2.0 

-0.4 

-3.0 

0.0 

3.0 

-0.5 

-3.5 

-0.5 

3.3 

-0.5 

-3.7 

-1.0 

3.5 

-0.3 

-3.5 

-1.7 

3.3 

0.0 

-3.0 

-2.2 

3.0 

0.6 

-2.5 

-2.3 

3.0 

1.0 

-1.0 

-2.1 

2.7 

1.3 

-0.5 

-1.9 

2.8 

1.6 

0.0 

-2.3 

2.5 

1.5 

0.5 

-2.4 

2.0 

1.4 

1.0 

-2.5 

1.0 

1.6 

0.8 

-1.6 

0.0 

1.5 

1.0 

-1.3 

6.  A  thermometer  is  found  to  have  the  following  errors 
when  calibrated  by  means  of  a  standard: 

10°+0.03 
12  -0.60 
18  -0.45 
22  -0.38 
30  +0.71 

Draw  a  curve  of  errors  for  the  thermometer. 


Determination  of  Constants. 
of  a  straight  line, 


Let  us  take  the  equation 


By  definition  -     is  the  tangent  of  the  angle  which  the 


PLOTTING 


57 


line  makes  with  the  z-axis.  This  is  equal  to  a  and  deter- 
mines this  constant.  When  x  =  Q,  y  =  b.  This  determines 
the  constant  6. 

Rearrangement  of  Data.  It  sometimes  happens  that 
if  a  curve  is  plotted  in  -one  way  it  is  meaningless  or  dif- 
ficult to  interpret;  while  a  rearrangement  of  data  makes 
it  perfectly  intelligible.  If,  for  example,  a  series  of  readings 
have  been  taken  with  a  Boyle's  law  apparatus,  the  result- 
ing curve  should  be  an  equilateral  hyperbola  (pv  =  a).  It 
will  be  found,  however,  that  the  readings  one  is  ordinarily 
able  to  obtain  are  not  sufficient  to  identify  the  curve.  If 
we  change  the  data  and  plot  v  and  l/p  we  get  a  straight  line. 

In  a  tangent  galvanometer  the  formula  is  I  =  k  tan  0. 
In  order  to  get  a  series  of  variations  in  the  current  we 
introduce  varying  resistances.  Since  I  =  E/R,  it  follows 
that  the  reciprocals  of  the  resistances  will  plot  a  straight 
line  with  tan  <j> 

EXERCISES 

1.  The  following  data  come  from  an  experiment  with 
the  Boyle's  law  apparatus: 


Volume. 

Pressure. 

Volume. 

Pressure. 

26.5 

134.3 

34.9 

102.9 

27.4 

130.5 

36.2 

99.3 

27.9 

126.4 

37.6 

95.7 

28.2 

122.9 

39.2 

92.3 

29.2 

118.4 

40.7 

88.3 

31.3 

114.5 

42.5 

85.3 

32.4 

110.5 

44.0 

81.9 

33.6 

106.7 

45.8 

78.9 

68 


THEORY  OF  MEASUREMENTS 


Discuss  the  accuracy  of  the  observations  from  the  plot. 

2.  The   following   table   is   made   up   from   resistances 

and  corresponding  deflections  with  a  tangent  galvanometer: 


Resistances  (Ohms). 

Deflections  (Degrees). 

90 

5.3 

50 

8.7 

30 

12.8 

20 

16.7 

10 

24.0 

5 

30.6 

3 

34.5 

1 

39.3 

0 

42.5 

a.  Assume  that  we  have  a  constant  electromotive  force 
and  make  a  plot  which  will  show  the  relation  between 
the  current  and  the  deflection. 

7  =  fctan0,  I  =  E/R. 

b.  Plot  the  resistances  with  the  cotangents  of  the  angles 
and  project  until  it  cuts  the  s-axis.    Interpret  the  curve. 

c.  Find  the  value  of  E  by  Ohm's  law. 

3.  Known  volumes  of  a  liquid  were  placed  in  a  flask 
and  weighed,  giving  data  as  follows: 


Volume  of  Liquid, 
cc. 

Total  Mass 
(Liquid  +Flask.) 
g. 

20.8 

204 

37.6 

221 

61.3 

250 

86.5 

287 

108.0 

307 

136.5 

336 

PLOTTING  59 

Plot  the  volume  on  the  z-axis  and  the  masses  on  the 


a.  Find  the  mass  of  the  flask  from  the  plot. 

6.  Find  the  specific  gravity  of  the  liquid. 

4.  Two  rulers  were  placed  together  at  random  and  the 
reading  on  the  centimeter  scale  was  taken  opposite  each 
inch  division  on  the  English  scale,  giving  data: 


Inches. 

Centimeters. 

0 

0 

1 

0 

2 

27.8 

3 

25.3 

4 

22.7 

5 

20.2 

6 

17.7 

7 

15.1 

8 

12.6 

9 

10.0 

10 

7.5 

11 

5.0 

12 

2.4 

Plot  the  inch  readings  on  the  z-axis  and  the  centimeter 
readings  on  the  y-axis. 

a.  Determine  from  the  plot  how  the  rulers  were  placed 
with  reference  to  each  other. 

6.  Find  the  ratio  of  the  centimeter  and  inch  by  using 
the  intercepts  of  the  curve  on  the  axes;  also  by  using  the 
ratio  of  the  difference  of  two  points  2/2—2/1  and  x%— x\. 

5.  The  candle-power  of  a  16-c.p.,  110-v.  lamp  was 
measured  for  different  voltages  across  the  terminals  as 
follows : 


60 


THEORY  OF  MEASUREMENTS 


Volts. 

Candle-power. 

60 

0.4 

70 

1.1 

80 

2.3 

90 

4.2 

100 

8.3 

105 

10.3 

118 

16.6 

128 

22.3 

138 

25.8 

150 

44.7 

160 

54.5 

170 

72.2 

180 

79.7 

190 

93.9 

200 

115.7 

210 

120.3 

Plot  volts  on  x  and  candle-powers  on  t/-axis. 

a.  Find  candle-power  at  110  v. 

6.  Find  candle-power  at  220  v. 

c.  Is  it  possible  to  estimate  the  voltage  at  which  the 
filament  first  begins  to  glow? 

6.  The  elongation  of  a  spring  was  measured  for  differ- 
ent loads,  and  the  energy  stored  in  the  spring  computed, 
giving  data: 


Loads. 

Elongation. 

Energy. 

g. 

cm. 

g.  cm. 

0 

0.0 

0 

50 

2.3 

57.5 

100 

5.4 

270.0 

150 

8.8 

660.0 

200 

11.9 

1190.0 

250 

15.2 

1900.0 

PLOTTING  61 

Plot  loads  (x)  with  elongations  (y)  and  energy  (y). 

a.  Compute  the  constant  of  the  spring  (g/cm). 

b.  We  have 


=  kL; 


Interpret  this  equation  and  compare  with  the  curve. 

c.  Compute  the  area  between  the  load-elongation 
curve  and  the  z-axis.  What  does  this  represent? 

The  electrical  resistance  of  lead  at  various  temperatures 
indicated  by  means  of  a  plot  that  the  resistance  would  be 
zero  at  —273°  C.  The  recent  experiments  of  Professor 
Kamerlingh  Onnes  have  justified  this  conclusion. 


CHAPTER  VII 


NEGLIGIBILITY 

Importance.  It  is  of  the  utmost  importance  that  stu- 
dents in  laboratory  courses  should  come  to  know  under 
just  what  conditions  they  are  at  liberty  to  neglect  small 
quantities  which  appear  in  their  work.  A  student  who 
comes  into  physics  from  courses  in  mathematics  is  con- 
scious of  a  decided  shock  when  he  is  told  to  throw  away 
certain  terms  in  an  equation.  It  is  hoped  that  a  few  illus- 
trations will  convince  him  that  such  a  process  is  not  at  all 
unscientific  or  inaccurate. 

We  learn  in  trigonometry  that  the  sine,  tangent,  and 
radian  value  of  an  angle  may  be  used  interchangeably  if 
the  angle  is  small.  The  following  table  will  make  this 
clear: 


Degrees. 

Radians. 

Sine. 

Tangent. 

0° 

0 

0 

0 

0   30' 

0.00873 

0.00873 

0.00873 

1 

0.01745 

0.01745 

0.01746 

2 

0.03491 

0.03490 

0.03492 

3 

0.05236 

0.05234 

0.05241 

4 

0.06981 

0.06976 

0.06993 

5 

0.08727 

0.08716 

0.08749 

62 


NEGLIGIBILITY  63 

It  will  be  seen  that  for  most  purposes  very  little  error 
would  result  from  substituting  the  radian  measure  or  the 
tangent  for  the  sine. 

EXERCISES 

1.  Construct  a  circle  with  radius  unity.  Take  an 
angle  of  60°  at  the  center  and  draw  an  arc  and  a  chord. 
Call  the  arc  dx  and  the  chord  dy  (equal  to  the  radius). 


Find  the  value  of  the  arc  by  radians     -^-,  and  the  chord 

by  the  law  of  sines.     In  this  case  dy=l  and  dx  =  1.0472. 
Find  relation  between  dx  and  dy  for  30°,  10°,  5°,  and  1°. 
2.  What  is  the  largest  angle  for  which  dx  =  dy  to  four 
places  of  decimals? 

The  Pendulum  Formula.  In  deducing  the  formula 
for  the  simple  pendulum,  we  make  use  of  an  approximation, 
so  that  the  equation  T=2irVl/g  is  not  quite  correct.  The 
equation 


tends  to  diminish  the  error  as  more  and  more  terms  are 

a 

used.     Here  K  =   -. 


EXERCISE 

Give  e  values  of  60°,  10°,  and  1°  and  assume  that  the 
longer  formula  gives  the  correct  value  of  g.  Find  the 
error  in  each  case  due  to  using  the  shorter  formula.  Re- 
member that  g  appears  as  a  square  root. 


64  THEORY  OF  MEASUREMENTS 

The  Mirror  Formula.  In  developing  the  formula  for 
the  circular  mirror  a  similar  approximation  is  noted.  By 
using  a  parabolic  mirror  no  approximation  appears. 

EXERCISES 

1.  Show  that  light  starting  from  the  focus  of  a  para- 
boloid of  revolution  will  go  in  parallel  lines  upon  reflection 
from  any  part  of  the  mirror. 

2.  Construct   a   parabola   whose   equation   is   y2  =  4px. 
Construct  a  circle  internally  tangent  to  this  whose  equa- 
tion is  x2-\-y2— 2rx=0.     Let  r  =  2p.     Select   points   along 
the  axis  distant  r/4,  r/2,   3r/4,   and  r  from  the  origin. 
Find  the  value  of  y  on  the  circle  and  the  corresponding 
value  of  y  on  the  parabola  in  each  case.    The  relation  of 
these  values  may  be  used  to  measure  the  error  due  to  the 
approximation. 

3.  Find  the  angular  aperture  at  the  center  of  the  circle 
for  each  point  taken. 

Approximate  Squares  and  Square  Roots.  There  is  a 
short  method  of  squaring  numbers  and  extracting  their 
square  roots  which  involves  an  approximation.  Let  us 
take  the  number  1.01  and  consider  it  made  up  of  1.0+0.01. 
If  we  square  this  by  use  of  the  binomial  theorem,  we  have 

1+0.02+0.0001  =  1.0201. 

This  is  nearly  equal  to  1.02.  If,  therefore,  we  square 
our  first  term  and  add  twice  the  second  term,  we  have 
a  rule  for  squaring  such  numbers  as  these.  The  square 
of  1.0019  is  1.0038036,  to  which  1.0038  is  a  close  approxi- 


NEGLIGIBILITY 


65 


mation.     If  the  whole  number  is  any  other  than  unity,  its 
value  must  be  multiplied  into  the  smaller  term. 

By  a  similar  process  we  may  find  the  square  root  of 
1.04  to  be  1.02,  and  that  of  4.04  to  be  2.01. 


The  Slide  Rule 

The  slide  rule  is  an  instrument  that  gives  approximate 
values  and  well  illustrates  the  principle  of  negligibility. 
The  following  table  shows  its  accuracy  for  certain  processes: 


Problem. 

Solution  by  Slide 
Rule. 

Solution  by  Five 
Place  Logs. 

Deviation, 

% 

152X391X31 

1842000 

1842400 

0.022 

86.4X0.028X4.95 

11.97 

11.975 

0.042 

496  -r-  381 

1.303 

1.3018 

0.21 

292-7-468 

0.624 

0.62394 

0.063 

25X0.07X1.151 

0.00543 

0.0054327 

0.05 

378X0.98       1 

The  Value  of  TT.  Much  energy  has  been  expended  in 
obtaining  values  of  TT  correct  to  a  large  number  of  decimal 
places.  It  has  been  carried  out  to  707  places.  (See  Seven 
Follies  of  Science  by  Phin.)  If  TT  is  carried  out  to  six 
or  seven  places,  it  is  sufficient  for  all  purposes.  The 
value  3^  will  answer  for  the  majority  of  cases.  It  will 
prove  convenient  to  remember  that  7r2  =  9.87  to  a  very  close 
approximation. 


66  THEORY  OF  MEASUREMENTS 

PROBLEMS 

1.  Find  the  error  due  to  using  the  value  3^  carried  out 
to  four  decimal  places,  for  TT. 

2.  Find  the  error  due  to  using  the  value  9.87  for  7r2, 
when  TT  is  carried  out  to  four  places. 

3.  We  are  accustomed  to  assume  in  physics  that  the 
coefficient  of  cubical  expansion  of  a  solid  is  three  times 
that  of  the  linear  expansion.     What  error  does  this  involve 
when  a  bar  of  brass,  100  cm.  long  at  0°  C.,  is  heated  to 
100°  C.? 

The  error  involved  in  using  a  mirror  and  scale  (Poggen- 
dorff's  method)  is  discussed  in  Stewart  and  Gee's  Practical 
Physics,  Vol.  I,  p.  55. 

CRITERIA  FOR  NEGLIBILITY 

It  often  happens  that  in  a  series  of  measurements  of 
the  same  quantity,  there  are  one  or  more  values  which  do 
not  compare  well  with  the  others.  The  question  arises, 
what  shall  be  done  with  these  observations?  They  seem 
to  have  been  taken  with  equal  care  with  the  others,  and 
there  seems  to  be  no  particular  reason  for  rejecting  them. 
Several  criteria  have  been  proposed  for  testing  such  obser- 
vations. 

The  Huge  Error.  We  find  the  mean  and  average 
deviation  (a.d.)  omitting  the  doubtful  observation;  then 
find  the  difference  between  the  doubtful  observation  and 
the  mean,  and  if  this  equals  or  is  greater  than  4  a.d.,  it 
should  be  rejected. 

Suppose  we  have  a  set  of  measurements:  10,  9,  10, 
9,  12.  The  last  measurement  is  questionable.  The  mean 


NEGLIGIBILITY 


67 


is  9.5.     a.d.=0.5.     12-9.5  =  2.5.    4X0.5  =  2.0.     The  meas- 
urement should  be  rejected. 

PROBLEM 

The  following  length  measurements  (centimeters)  have 
been  taken:  60.1,  60.2,  60.3,  59.9,  58.3,  63.1.  Apply  the 
method  to  the  last  two  results. 

Chauvenet's  Criterion.  This  is  a  more  elaborate 
method.  We  may  let  t  stand  for  the  ratio  between  the 
limiting  error  (x)  and  the  probable  error  of  a  single  observa- 
tion (r).  If  there  are  n  errors,  we  may  assume  that  nP 
will  be  the  number  less  than  x,  and  n—nP  the  number 
greater  than  x. 

Then,  by  definition 


2n-l 
2n    ' 


We  have 


2     f* 
=—T=  I  e~*dt.    (See  page  15.) 

VTT^0 


If  we  equate  these  two  values  of  P  and  solve,  we  may  find 
a  value  of  t  corresponding  to  any  value  of  n.  A  few 
common  values  of  n  are  given: 


n 

t 

n 

t 

n 

l 

3 

2.05 

7 

2.67 

11 

2.96 

4 

2.27 

8 

2.76 

12 

3.02 

5 

2.44 

9 

2.84 

13 

3.07 

6 

2.57 

10 

2.91 

14 

3.12 

68  THEORY  OF  MEASUREMENTS 

Since  the  limiting  error,  x  =  tr,  we  have  only  to  find 
these  two  values  and  compare. 

Let  us  take  the  following  data:  12,  13,  12,  13,  13,  15. 
The  mean  is  13  and 

S^  =  6.    r  =  0.6745  \—,=  0.6745x/f  =  0.74. 

\  71 —  1  \  O 

For  n  =  6,    £=2.57,    x  =  tr  =  l&. 

The  deviation  of  the  term  in  question  is  2.0  and  it  should 
be  rejected. 

PROBLEM 
The  following  observations  were  taken  with  a  transit, 

43°  52'  26".4 
28  .5 

27  .8 

28  .0 
31  .3 
30  .2 
27  .9 

Test  such  observations  as  may  be  necessary  by  Chau- 
venet's  Criterion. 

Criterion  for  Negligibility  for  Deviation  in  Components 

When  the  deviation  in  any  component  contributes 
but  little  to  the  deviation  in  the  result,  it  is  sometimes 
proper  to  neglect  it  altogether.  We  know  that  the  error 
in  the  final  result, 


.  .  .  AB2. 


NEGLIGIBILITY  69 

Suppose  A2  is  the  quantity  under  investigation.     Let 


.  .  .  An2, 

where  A22  has  been  omitted. 

A— Ao  =  the  reduction  in  the  deviation  in  the  result 
due  to  omitting  A2.  We  may  assume  this  value  to  be  any- 
thing we  please,  depending  upon  the  accuracy  we  require. 
If  we  insist,  as  some  authors  suggest,  that  it  be  equal  to 
or  less  than  TVA,  we  have 

A0f0.9A. 

But      A22  =  A2-A02  =  A2(1-0.92)=0.19A2. 
A2  =  0.43A. 

We  may  thus  neglect  a  deviation  if  it  contributes  less  than 
0.43  of  the  total  deviation  of  the  result.  A  more  rigid 
criterion  could,  of  course,  be  determined  by  substituting 
a  smaller  value  for  the  TV 

Significant  Figures.  This  is  a  subject  that  may  be 
properly  treated  under  the  head  of  negligibility.  At  each 
stage  in  a  series  of  measurements  the  student  should  look 
over  his  work  and  eliminate  such  figures  as  lend  nothing 
to  its  precision.  A  rather  full  treatment  of  this  subject 
is  given  in  Holman's  Precision  of  Measurements,  to  which 
the  student  is  referred. 

The  following  rules  of  precision  are  given: 

1.  When  a  rejected  figure  is  five  or  over,  increase  the 
previous  figure  by  one. 

If  we  decide  to  drop  the  7  in  14.637,  we  should  write 
it  14.64. 


70  THEORY  OF  MEASUREMENTS 

2.  In   the   deviation   measure,    we   should   retain   two 
significant  figures.     Thus  a.d.  =  0.062 ;  r  =  1 .6. 

3.  In  our  measured  quantity,  we  should  retain  places 
corresponding  to  the  second  significant  figure  in  the  devia- 
tion measure. 

m  =  368.731±0.21  becomes  368.73, 
ra  =  406.67±2.6  becomes  406.7. 

4.  In  adding  quantities,   retain  such  places  as  corre- 
spond to  the  number  having  the  largest  deviation  measure. 

Quantity.  a.d.  Result. 

374.20        0.36         374.20 

4768.121       0.012       4768.12 

984.1698      0.0021       984.17 

6126.49 

5.  In   multiplication   and   division,   find   the   quantity 
whose  -  -  is  the  greatest.     From  this  find  the  percentage 

precision  and  if  this  is  1  per  cent  or  more,  use  four  significant 
figures;  if  between  1  and  0.1  per  cent  use  five  significant 
figures;  if  between  0.1  and  0.01  per  cent,  use  six  significant 
figures. 

If  we   have   to   multiply  41.6±0.4   by   590.25±0.06, 

we  note  that  — —  is  greater  for  the  first  than  for  the  second. 
m 

As  this  is  less  than  1  per  cent,  we  should  retain  five  sig- 
nificant figures,  giving  24556  as  a  result. 


NEGLIGIBILITY  71 

PROBLEMS 

1.  We  have  given  the  number  504.628  with  the  fol- 
lowing   deviation    measures:     a.d.  =  0.21;     r  =  0.031;     cor- 
rect to  one  part  in  one  hundred;   correct  to  two  per  cent. 
What  is  the  proper  expression  for  the  number  in  each  case? 

2.  Add 

21.42  dhO.61 
338. 161  ±0.042 
543.1     ±1.5 

3.  Multiply 

630.45±0.62  by  25.635 ±0.024. 


CHAPTER  VIII 
EMPIRICAL  FORMULAE  AND  CONSTANTS 

Definitions.  A  mathematical  formula  is  one  that  is  de- 
duced by  a  process  of  reasoning  along  mathematical  lines. 
The  formula  for  the  distance  passed  over  ;by  a  falling  body  is 
an  illustration. 

S  =  v0t+%at2  is  derived  from  certain  definitions  of 
velocity  and  acceleration,  to  which  mathematical  processes 
have  been  applied. 

An  empirical  formula  cannot  be  derived  in  this  man- 
ner, but  depends  upon  the  results  of  experiments  which 
are  treated  in  a  manner  to  be  described.  The  following 
is  a  typical  empirical  formula: 

0  =  980.6056-2.5028  cos  2Z-O.OOOOOS*. 

This  has  been  made  up  from  a  large  number  of  measure- 
ments of  g  under  various  conditions  of  latitude  (I)  and 
altitude  (h). 

In  order  to  explain  the  method  employed  in  constructing 
empirical  formulae,  let  us  take  an  experiment  illustrating 
the  relation  of  the  space  passed  over  by  a  falling  body, 

to  the  time  of  fall. 

72 


EMPIRICAL  FORMULAE  AND  CONSTANTS  73 

The  following  may  be  assumed  to  be  the  results  of  the 
experiment: 

t  9 

1  1 

2  4 

3  9 

4  16 

The  obvious  relation  existing  between  t  and  s  will  be 
disregarded,   and  we  will  endeavor  to  find  an  empirical 


FIG.  8. 

formula  which  fits  the  conditions.     Our  first  step  is  to 
plot  a  curve. 

This  is  seen  to  have  the  characteristics  of  a  parabola 
and  we  write  the  general  equation, 

y  =  S+Tx+Ux2+,  etc. 

x  has  been  substituted  for  t,  and  y  for  s.     From  this  are 
found  observation  equations: 


74  THEOKY  OF  MEASUREMENTS 

1=S+T+U 


The  normal  equations  are: 


30=4S+10!T+30t7 
100  =  10/S+307M-  100*7 


When  these  equations  are  solved,  we  have 


and  our  equation  becomes  y=x2. 

If  the  experimenter  had  used  only  three  seconds  and 
had  recorded  an  8  instead  of  a  9,  the  values  would  have 
been 

17  =  0.5 


Giving  y=  -  1+1.5*  -f0.5z2. 

Upon  repeating  the  experiment  as  the  space  measurement 
approaches  closer  to  9  this  formula  approaches  the  value 
!/=**. 


EMPIRICAL  FORMULAE  AND  CONSTANTS  75 

While  no  one  would  think  of  actually  applying  this 
method  to  so  simple  a  case,  it  affords  a  good  illustration 
of  the  manner  in  which  all  empirical  formulae  are  con- 
structed. 

Classes  of  Curves.  The  following  are  some  of  the 
common  curves  from  which  these  formulae  may  be  con- 
structed, together  with  their  formulae  and  illustrations: 

1.  Straight  line.    x  =  y. 

2.  Parabolic.         y  =  S+Tx+Ux  +,  etc. 

3.  Cyclic.  y  =  S+T  sin  — x+Tr  cos  — x+,  etc. 

4.  Logarithmic.     y  =  beax. 

5.  Hyperbolic.    xy  =  a. 

The  formula  for  velocity  due  to  gravity,  v  =  gt,  illus- 
trates the  first  class.  The  space  passed  over  by  a  falling 
body,  the  change  in  velocity  of  a  river  below  its  surface, 
and  the  growth  of  the  United  States  in  population,  illustrate 
the  second  class.  Cyclic  curves  may  be  used  to  represent 
the  rise  and  fall  of  temperature,  pressure,  and  humidity 
through  a  given  interval.  The  logarithmic  curve  represents 
plots  made  from  Newton's  law  of  cooling,  the  absorption 
of  light  for  varying  thicknesses,  and  the  gain  due  to  com- 
pound interest.  A  familiar  example  of  a  hyperbolic  curve 
is  afforded  by  Boyle's  law,  pv  =  ct.  This  will  plot  a  rec- 
tangular hyperbola. 

Rules.  The  method  of  procedure  may  be  summarized 
in  the  following  rules: 

1.  Write  the  observation  equation. 

2.  Plot  the  curve. 


76  THEORY  OF  MEASUREMENTS 

3.  Identify  the  curve  and  write  its  equation. 

4.  Form  the  normal  equations. 

5.  Solve  for  the  unknowns,  and  these  give  the  values 
of  the  constants  sought. 

It  will  be  observed  that  every  additional  observation 
taken  alters  the  values  of  the  constants. 

EXERCISES  AND  PROBLEMS 

For  the  straight  line  and  rectangular  hyperbola  an 
inspection  of  the  data  is  generally  sufficient  to  determine 
the  value  of  the  constant. 

For  the  parabola  the  following  examples  should  be 
studied: 


1.  Vertical  velocity  curve  on  the  Mississippi  river. 

Depth.  Velocity. 

0 . 0  foot  3 . 1950  feet  per  second 

0.1  3.2299 

0.2  3.2532 

0.3  3.2611 

0.4  3.2516 

0.5  3.2282 

0.6  3.1807 

0.7  3.1266 

0.8  3.0594 

0.9  2.9759 

The  curve  is  obviously  a  parabola. 


EMPIRICAL  FORMULAE  AND  CONSTANTS          77 
The  first  two  observation  equations  are 


Write  down  the  remaining  eight  observation  equations, 
form  the  normals,  and  solve.    The  results  are 

S  =  3.195  13 

7=0.44253 

U  =-0.7653,  giving  the  formula 

F  =  3.19513+0.44253X-0.7653X2. 


FIG.  9. 

It  will  prove  of  interest  to  determine  the  velocity  for 
each  depth  and  compare  with  that  obtained  by  experiment. 
(Why  should  they  fail  to  check?) 

2.  The  growth  in  the  population  of  the  United  States 
is  a  good  illustration  of  this  curve,  although  from  the  nature 
of  the  case  the  results  are  not  very  reliable.  In  Popular 
Science  Monthly  for  April,  1910,  page  382,  a  set  of  curves 
are  drawn  for  various  countries.  It  will  be  seen  that  Sweden 
and  Norway,  Turkey,  Spain,  and  Italy  may  be  fairly  rep- 
resented by  straight  lines.  The  United  States  shows  an 


78 


THEORY  OF  MEASUREMENTS 


easily  recognized  parabola.     In  the  article  referred  to  the 
complete  solution  of  the  census  problem  is  given. 
3.  The  following  illustrates  a  cyclic  curve: 


e 

Y 

e 

Y 

0° 

20 

200° 

14 

20 

38 

220 

-13 

40 

51 

240 

-24 

60 

64 

260 

-30 

80 

68 

280 

-28 

100 

70 

300 

-24 

120 

64 

320 

-13 

140 

53 

340 

-  2 

160 

16 

360 

20 

180 

20 

Substitute  6  for  ~#  in  the  equation  and  plot  the  curve. 


m 


Use  the  first  three  terms  and  form  an  observation  equation 
for  each  value  of  9.  We  have  for  the  first  four  observa- 
tion equations: 


20=/S-r-Trsin 

sin  20°+!T'  cos  20° 
sin  40°+r  cos  40° 
sin  60°+r  cos  60° 


Normal  equations  for  S,  T,  and  T'  are  formed  from  these 
and  their  solution  supplies  the  values  of  the  constants  in 
the  formula. 

4.  We  may  use  Winkehnann's  data  showing  the  rela- 


EMPIRICAL  FORMULAE  AND  CONSTANTS  79 

tion  between  the  temperature  of  a  cooling  body  at  differ- 
ent times,  as  an  illustration  of  a  logarithmic  curve. 

o  ti  -h 

18.9  3.45 

X6.9  10.85 

14.9  19.30 

12.9  28.8(0 

10.9  40.10 

8.9  53.75 

6.9  >0.95 

The  logarithmic  relations  of  these  quantities  is  not 
obvious  by  direct  inspection.  It  is  reached  by  the  fol- 
lowing process:  We  assume  that  the  ratio  at  which  a  body 
loses  heat  (—  dQ)  is  proportional  to  the  difference  between 
its  temperature  and  that  of  its  surroundings.  This  is  ex- 
pressed by 


By  definition  of  specific  heat 


Substitute  this  in  the  equation  above 

dQ  k 

—  —  -=a0,   where   a=-    and    00=0. 

From  this  we  get  by  integrating 

log  b—  log  0=at. 


€0  -THEORY  OF  MEASUREMENTS 

Now  if  0i  represents  temperature  at  time  ti,  and  62  tem- 
perature at  time  fe,  we  have 


log  6—  log  02  =  afe; 

1  A 

From  which  a  =  -  —  —  log  •=-> 


Take  02  =  19.9  and  substitute  the  values  of  0  for  0i 
and  show  that  a  is  a  constant,  approximately  equal  to 
0.0065. 

Such  an  equation  as  ~"Jt=a^t  where  the  change  in  a 

quantity  is  proportional  to  the  quantity  itself,  is  called 
a  "  compound  interest  equation/' 


INDEX 


Adjustment  of  Observations 22 

Approximate  Squares  and  Square  Roots 64 

Arithmetical  Mean 17 

Average  Deviation 38 

Best  Magnitudes  and  Best  Ratios 51 

Chauvenet's  Criterion 67 

Constant  Interval 17 

Constants  in  an  Equation 56 

Empirical  Formulae  and  Constants 72 

Errors — Classes  of 3 

Fractional  Method 50 

Huge  Error 66 

Least  Squares 16 

Mean  Square  Error 37 

Measurements — Classes  of 2 

Negligibility 62 

Normal  Equations 24 

Plotting 54 

Precision  of  Observations 36 

Probable  Error 38 

Probability 4 

Probability  Curve 9 

Probability  Integral 14 

Propagation  of  Errors 44 

Propagation  of  Errors — Converse  Problem 49 

Residuals 12 

Short  Methods  of  Adjustment 32 

Significant  Figures 69 

Slide  Rule 65 

Value  of  T 65 

Weighting 18 

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Bernthsen,  A.     A  Text-book  of  Organic  Chemistry.     Trans,  by 

G.  M'Gowan i2mo,     *2  50 

Berry,  W.  J.     Differential  Equations  of  the  First  Species. 

I2mo  (In  Preparation.) 
Bersch,  J.     Manufacture  of  Mineral  and  Lake  Pigments.     Trans. 

by  A.  C.  Wright 8vo,     *s  oo 


6      D.    VAN    NOSTRAND    COMPANY'S    SHORT-TITLE    CATALOG 

Bertin,  L.  E.     Marine  Boilers.     Trans,  by  L.  S.  Robertson .  .8vo, 

Beveridge,  J.     Papermaker's  Pocket  Book i2ino, 

Binnie,  Sir  A.     Rainfall  Reservoirs  and  Water  Supply.  .8vo, 

Binns,  r    F.     Manual  of  Practical  Potting 8vo,    *7 

The  Potter's  Craft i2mo,    *; 

Birchmore,  W.  H.    Interpretation  of  Gas  Analysis i2mo,    *i 

Blaine,  R.  G.    The  Calculus  and  Its  Applications i2mo,    *• 

Blake,  W.  H.     Brewer's  Vade  Mecum 8vo, 

Blasdale,   W.    C.     Quantitative    Chemical   Analysis.  ..  .i2mo,    *2 
Bligh,  W.  G.    The  Practical  Design  of  Irrigation  Works.  .8vo,     *6 

Bloch,  L.     Science  of  Illumination 8vo,    *2 

Blok,  A.    Illumination  and  Artificial  Lighting i2mo,     *i 

Blucher,  H.     Modern  Industrial  Chemistry.     Trans,  by  J.  P. 

Millington    8vo,     *7 

Blyth,  A.  W.    Foods:  Their  Composition  and  Analysis.  ..8vo,      7 

Poisons:    Their  Effects  and  Detection 8vo,      7 

Bockmann,  F.     Celluloid i2mo, 

Bodmer,  G.  R.    Hydraulic  Motors  and  Turbines i2mo, 

Boileau,  J.  T.  Traverse  Tables 8vo, 

Bonney,  G.  E.     The  Electro-plater's  Handbook i2mo, 

Booth,  N.    Guide  to  Ring-Spinning  Frame i2rno,    *i 

Booth,  W.  H.    Water  Softening  and  Treatment 8vo,      *2 

Superheaters  and  Superheating  and  their  Control.  .  .8vo,    *i 

Bottcher,  A.    Cranes:  Their  Construction,  Mechanical  Equip- 
ment and  Working.    Trans,  by  A.  Tolhausen. ..  .410,  *io 
Bottler,  M.     Modern  Bleaching  Agents.    Trans,  by  C.  Salter. 

i2mo,    *2 

Bottone,  S.  R.     Magnetos  for  Automobilists i2mo,    *i 

Boulton,  S.  B.    Preservation  of  Timber.     (Science  Series  No. 

82.)     i6mo,      o 

Bourcart,  E.    Insecticides,  Fungicides  and  Weedkillers . . .  8vo,     *4 
Bourgougnon,  A.    Physical  Problems.  (Science  Series  No.  113.) 

i6mo,      o 
Bourry,    E.      Treatise    on    Ceramic    Industries.      Trans,    by 

A.  B.  Searle     8vo,     *s 

Bowie,  A.  J.,  Jr.  A  Practical  Treatise  on  Hydraulic  Mining. 8vo,       5 
Bowles,  0.  Tables  of  Common  Rocks.    (Science  Series.)  .i6mo,      o 


D.    VAN    NOSTRAND    COMPANY^   SHORT-TITLE    CATALOG     7 

Bowser,  E.  A.  Elementary  Treatise  on  Analytic  Geometry. tamo,  i  75 
—  Elementary   Treatise   on   the    Differential    and   Integral 

Calculus    i2mo,  2  25 

Bowser,  E.  A.  .Elementary  Treatise  on  Analytic  Mechanics, 

i2mo,  3  oo 

Elementary   Treatise   on   Hydro-mechanics .i2mo,  2  50 

A  Treatise  on  Roofs  and  Bridges *2  25 

Boycott,  G.  W.  M.     Compressed  Air  Work  and  Diving.  .8vo,  *4  oo 

Bragg,   E.  M.     Marine   Engine  Design i2mo,  *2  oo 

Design  of  Marine  Engines  and  Auxiliaries (In  Press.) 

Brainard,  F.  R.    The  Sextant.    (Science  Series  No.  ioi.).i6mo, 

Brassey's  Naval  Annual  for  1911 8vo,  *6  oo 

Brew,  W.     Three-Phase  Transmission 8vo,  *2  oo 

Briggs,  R.,  and  Wolff,  A.  R.    Steam-Heating.    (Science  Series 

No.   67.)    i6mo,  o  50 

Bright,  C.    The  Life  Story  of  Sir  Charles  Tilson  Bright.  .8vo,  *4  50 
Brislee,  T.  J.    Introduction  to  the  Study  of  Fuel.     (Outlines 

of  Industrial  Chemistry.) 8vo,  *s  oo 

Broadfoot,  S.  K.    Motors  Secondary  Batteries.     (Installation 

Manuals  Series.)    121110,  *o  75 

Broughton,  H.  H.    Electric  Cranes  and  Hoists *9  oo 

Brown,  G.  Healthy  Foundations.   (Science  Series  No.  80.). i6mo,  o  50 

Brown,  H.     Irrigation 8vo,  *s  oo 

Brown,  Wm.  N.    The  Art  of  Enamelling  on  Metal i2mo,  *i  oo 

Handbook  on  Japanning  and  Enamelling i2mo,  *i  50 

House   Decorating  and   Painting i2mo,  *i  50 

History   of   Decorative   Art i2mo,  *i  25 

Dipping,    Burnishing,    Lacquering    and    Bronzing    Brass 

Ware i2mo,  *x  oo 

Workshop  Wrinkles 8vo,  *i  oo 

Browne,  R.  E.    Water  Meters.    (Science  Series  No.  8i.).i6mo,  o  50 

Bruce,  E.  M.     Pure  Food  Tests i2mo,  *i  25 

Bruhns,  Dr.     New  Manual  of  Logarithms 8vo,  cloth,  2  oo 

Half  morocco,  2  50 
Brunner,  R.  Manufacture  of  Lubricants,  Shoe  Polishes  and 

Leather  Dressings.     Trans,  by  C.  Salter 8vo,  *s  oo 


8       D.    VAN    NOSTRAND    COMPANY'S    SHORT-TITLE    CATALOG 

Buel,  R.  H.    Safety  Valves.     (Science  Series  No.  21.)  .  .  ,i6mo,  o 
Burley,    G.    W.     Lathes,    Their   Construction    and    Operation, 

i2ino,  i 

Burstall,  F.  W.     Energy  Diagram  for  Gas.     With  text.  .  .8vo,  *i 

—  Diagram  sold  separately • *i 

Burt,  W.  A.     Key  to  the  Solar  Compass i6mo,  leather,  2 

Buskett,  E.   W.     Fire   Assaying i2mo,  *i 

Butler,  H.  J.    Motor  Bodies  and  Chasis 8vo,  *2 

Byers,    H.    G.,    and    Knight,    H.    G.      Notes    on    Qualitative 

Analysis    8vo,    *i 

Cain,  W.    Brief  Course  in  the  Calculus i2mo, 

—  Elastic  Arches.      (Science  Series  No.  48.) i6mo, 

—  Maximum  Stresses.     (Science  Series  No.  38.) i6mo, 

—  Practical  Dsigning  Retaining  of  Walls.    (Science  Series 

No.  3.)    i6mo,      o 

—  Theory  of  Steel-concrete  Arches  and  of  Vaulte'd  Struc- 

tures.    (Science  Series.) i6mo,      o 

—  Theory  of   Voussoir  Arches.      (Science   Series  No.    12.) 

i6mo,      o 

—  Symbolic  Algebra.     (Science  Series  No.  73.) i6mo,      o 

Carpenter,  F.  D.     Geographical  Surveying.     (Science  Series 

No.  37.)    i6mo, 

Carpenter,  R.  C.,  and  Diederichs,  H.  Internal-Combustion 

Engines  8vo,  *$ 

Carter,  E.  T.  Motive  Power  and  Gearing  for  Electrical  Ma- 
chinery   8vo, 

Carter,  H.  A.    Ramie  (Rhea),  China  Grass i2mo, 

Carter,  H.  R.  Modern  Flax,  Hemp,  and  Jute  Spinning. .  8vo, 
—  Bleaching,  Dyeing  and  Finishing  of  Fabrics 3vo,  *: 

Cary,  E.  R.  Solution  of  Railroad  Problems  With  the  Use  of 

the  Slide  Rule i6mo,  *: 

Cathcart,  W.  L.    Machine  Design.    Part  I.   Fastenings .  . .  8vo,    *; 

Cathcart,  W.  L.,  and  Chaffee,  J.  I.  Elements  of  Graphic 

Statics  and  General  Graphic  Methods 8vo,  *< 

Short  Course  in  Graphic  Statics i2mo,    *] 


D.    VAN    NOSTRAND    COMPANY^    SHORT-TITLE    CATALOG      9 

Caven,  R.  M.,  and  Lander,  G.  D.    Systematic  Inorganic  Chem- 
istry       I2H10,  *2    00 

Chalkley,  A.  P.     Diesel  Engines 8vo,  *3  oo 

Chambers'  Mathematical  Tables 8vo,  i  75 

Chambers,  G.  F.    Astronomy 121110,  *i  50 

Charpentier,    P.     Timber 8vo,  *6  oo 

Chatley,  H.    Principles  and  Designs  of  Aeroplanes.    (Science 

Series.)    i6mo,  o  50 

How  to  Use  Water  Power lamo,  *i  oo 

Child,   C.  D.     Electric   Arcs 8vo,  *2  oo 

Child,  C.  T.    The  How  and  Why  of  Electricity i2mo,  i  oo 

Christian,  M.    Disinfection  and  Disinfectants i2mo,  *2  oo 

Christie,   W.   W.     Boiler-waters,   Scale,   Corrosion,  Foaming, 

8vo,  *s  oo 

Chimney  Design  and  Theory 8vo,  *s  oo 

Furnace    Draft.      (Science    Series.) i6mo,  o  50 

Water,  Its  Purification  and  Use  in  the  Industries.  .8vo, 

Church's  Laboratory  Guide.    Rewritten  by  Edward  Kinch .  8vo,  *2  50 

Clapperton,  G.     Practical   Papermaking 8vo,  250 

Clark,  A.  G.     Motor  Car  Engineering. 

Vol.   I.     Construction    8vo,  *s  oo 

Vol.  II.     Design (In   Press.) 

Clark,  C.  H.     Marine  Gas  Engines i2mo,  *i  50 

Clark,  J.  M.    New  System  of  Laying  Out  Railway  Turnouts, 

i2ino,  i  oo 
Clarke,  J.  W.,  and  Scott,  W.    Plumbing  Practice. 

Vol.      I.    Lead  Working  and  Plumbers'  Materials . .  8vo,  *4  oo 

Vol.    II.    Sanitary  Plumbing  and  Fittings '. . .  (In  Press.) 

Vol.  III.    Practical  Lead  Working  on  Roofs (In  Press.) 

Clerk,    D.,    and    Idell,    F.    E.      Theory    of    the    Gas    Engine. 

(Science  Series  No.  62.) i6mo,  o  50 

Clevenger,   S.   R.     Treatise   on   the  Method   of   Government 

Surveying   i6mo,  mor.,  2  50 

Clouth,  F.     Rubber,  Gutta-Percha,  and  Balata 8vo,  *$  oo 

Cochran,  J.    Treatise  on  Cement  Specifications 8vo,  *i  oo 

Concrete  and  Reinforced  Concrete  Specifications 8vo,  *2  50 


10  D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Coffin,  J.  H.  C.     Navigation  and  Nautical  Astronomy. .  i2mo,  *3  50 
Colburn,  Z.,  and  Thurston,  R.  H.     Steam  Boiler  Explosions. 

(Science  Series  No.   2.) i6mo,  o  50 

Cole,  R.  S.     Treatise  on  Photographic  Optics i2mo,  i  50 

Coles-Finch,  W.     Water,  Its  Origin  and  Use 8vo,  *5  oo 

Collins,  J.  E.     Useful  Alloys  and  Memoranda  for  Goldsmiths, 

Jewelers , i6mo,  o  50 

Collis,  A.  G.    High  and  Low  Tension  Switch-Gear  Design .  8vo,  *3  50 

Switchgear.     (Installation  Manuals  Series.) i2mo,  o  50 

Coombs,  H.  A.     Gear  Teeth.     (Science  Series  No.  120). . .  i6mo,  o  50 

Cooper,  W.  R.     Primary  Batteries 8vo,  *4  oo 

Copperthwaite,  W.  C.     Tunnel  Shields 4to,  "9  oo 

Corey,  H.  T.     Water  Supply  Engineering 8vo  (In  Press.) 

Corfield,  W.  H.  Dwelling  Houses.  (Science  Series  No.  50.)  i6mo,  o  50 

Water  and  Water-Supply.     (Science  Series  No.  17.). .  i6mo,  j  50 

Cornwall,  H.  B.     Manual  of  Blow-pipe  Analysis 8vo,  *2  50 

Cowell,  W.  B.     Pure  Air,  Ozone,  and  Water i2mo,  *2  oo 

Craig,  J.  W.,  and  Woodward,  W.  P.    Questions  and  Answers 

about  Electrical  Apparatus i2mo,  leather,  i  50 

Craig,  T.     Motion  of  a  Solid  in  a  Fuel.     (Science  Series  No.  49.) 

i6mo,  o  50 

Wave  and  Vortex  Motion.     (Science  Series  No.  43.) .  i6mo,  o  50 

Cramp,  W.     Continuous  Current  Machine  Design 8vo,  *2  50 

Greedy,  F.  Single-Phase  Commutator  Motors 8vo,  *2  oo 

Crocker,  F.  B.     Electric  Lighting.     Two  Volumes.     8vo. 

Vol.   I.     The  Generating  Plant 3  oo 

Vol.  II.    Distributing  Systems  and  Lamps 

Crocker,  F  B.,  and  Arendt,  M.     Electric  Motors 8vo,  *2  50 

and  Wheeler,  S.  S.    The  Management  of  Electrical  Ma- 
chinery   I2H10,  *I   OO 

Cross,  C.  F.,  Bevan,  E.  J.,  and  Sindall,  R.  W.     Wood  Pulp  and 

Its  Applications.     (Westminster  Series.) 8vo,  *2  oo 

Crosskey,  L.  R.     Elementary  Prospective 8vo,  i  oo 

Crosskey,  L.  R.,  and  Thaw,  J.     Advanced  Perspective 8vo,  i  50 

Culley,  J.  L.    Theory  of  Arches.    (Science  Series  No.  87.) .  i6mo,  o  50 

Dadourian,  H.  M.    Analytical  Mechanics 8vo.  *3  oo 

Danby,  A.    Natural  Rock  Asphalts  and  Bitumens 8vo,  *a  50 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG    11 

Davenport,  C.     The  Book.     (Westminster  Series.) 8vo,  *2  oo 

Davey,  N.     The  Gas  Turbine 8vo,  *4  oo 

Da  vies,  F.  H.      Electric  Power  and  Traction 8vo,  *2  oo 

Foundations  and  Machinery  Fixing.   (Installation  Manuals 

Series.) i6mo,  i  oo 

Dawson,  P.     Electric  Traction  on  Railways 8vo,  *Q  oo 

Deerr,  N.     Cane  Sugar 8vo,  7  oo 

Deite,  C.     Manual  of  Soapmaking.     Trans,  by  S.  T.  King.  -4to,  *5  oo 
De  la  Coux,  H.     The  Industrial  Uses  of  Water.     Trans,  by  A. 

Morris 8vo,  *4  50 

Del  Mar,  W.  A.     Electric  Power  Conductors 8vo,  *2  oo 

Denny,  G.  A.     Deep-Level  Mines  of  the  Rand 4to,  *io  oo 

• Diamond  Drilling  for  Gold *5  oo 

De  Roos,  J.  D.  C.     Linkages.     (Science  Series  No.  47.). . .  i6mo,  o  50 

Derr,  W.  L.     Block  Signal  Operation Oblong  i2mo,  *i  50 

Maintenance  of  Way  Engineering (In  Preparation.) 

Desaint,  A.     Three  Hundred  Shades  and  How  to  Mix  Them. 

8vo,  8  oo 

De  Varona,  A.     Sewer  Gases.     (Science  Series  No.  55.)...  i6mo,  050 
Devey,  R.  G.     Mill  and  Factory  Wiring.     (Installation  Manuals 

Series.) izmo,  *i  oo 

Dibdin,  W.  J.     Purification  of  Sewage  and  Water .8vo,  6  50 

Dichman,  C.    Basic  Open-Hearth  Steel  Process 8vo,  *3  50 

Dieterich,  K.    Analysis  of  Resins,  Balsams,  and  Gum  Resins 

8vo,  *s  oo 
Dinger,  Lieut.  H.  C.     Care  and  Operation  of  Naval  Machinery 

I2IT1O.  *2    OO 

Dixon,  D.  B.     Machinist's  and  Steam  Engineer's  Practical  Cal- 
culator   i6mo,  mor.,  i  25 

Doble,  W.  A.    Power  Plant  Construction  on  the  Pacific  Coast.  (In  Press.) 

Dommett,  W.  E.    Motor  Car  Mechanism zarno,  *i  25 

Dorr,  B.  F.     The  Surveyor's  Guide  and  Pocket  Table-book. 

i6mo,  mor.,  2  oo 

Draper,   C.   H.     Elementary   Text-book   of   Light,    Heat   and 

Sound i2mo,  i  oo 

Draper,  C.  H.      Heat  and  the  Principles  of  Thermo-dynamics, 

New  and  Revised  Edition i2ino,  2  oo 


12    D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Dron,  R.  W.  Mining  Formulas i2mo,  i  oo 

Dubbel,  H.  High  Power  Gas  Engines 8vo,  *5  oo 

Duckwall,  E.  W.  Canning  and  Preserving  of  Food  Products, 8 vo,  *5  oo 
Dumesny,  P.,  and  Noyer,  J.  Wood  Products,  Distillates,  and 

Extracts . .  8vo,  *4  50 

Duncan,  W.  G.,  and  Penman,  D.  The  Electrical  Equipment  of 

Collieries 8 vo,  *4  oo 

Dunstan,  A.  E.,  and  Thole,  F.  T.  B.  Textbook  of  Practical 

Chemistry i2mo,  *i  40 

Duthie,  A.  L.  Decorative  Glass  Processes.  (Westminster 

Series) 8vo,  *2  oo 

Dwight,  H.  B.  Transmission  Line  Formulas 8vo,  *2  oo 

Dyson,  S.  S.  Practical  Testing  of  Raw  Materials 8vo,  *5  oo 

—  and  Clarkson,  S.  S.    Chemical  Works 8vo,  *7  30 

Eccles,  W.  H.    Wireless  Telegraphy  and  Telephony (In  Press.) 

Eck,  J.     Light,  Radiation  and  Illumination.    Trans,  by  Paul 

Hogner 8vo,  *2  50 

Eddy,  H.  T.     Maximum  Stresses  under  Concentrated  Loads, 

8vo,  i  50 

Edelman,  P.     Inventions  and  Patents i2mo,   (In  Press.) 

Edgcumbe,  K.     Industrial  Electrical  Measuring  Instruments . 

8vo. 
Edler,  R.    Switches  and  Switchgear.    Trans,  by  Ph.  Laubach. 

8vo,  *4  oo 

Eissler,  M."  The  Metallurgy  of  Gold 8vo,  7  50 

—  The  Metallurgy  of  Silver 8vo,  4  oo 

—  The  Metallurgy  of  Argentiferous  Lead 8vo,  5  oo 

—  A  Handbook  of  Modern  Explosives 8vo,  5  oo 

Ekin,  T.  C.      Water  Pipe  and    Sewage    Discharge  Diagrams 

folio,  *3  oo 

Electric  Light  Carbons,  Manufacture  of 8vo,  i  oo 

Eliot,  C.  W.,  and  Storer,  F.  H.    Compendious  Manual  of  Qualita- 
tive Chemical  Analysis I2mo,  *i  25 

Ellis,  C.     Hydrogenation  of  Oils 8vo,  *4  oo 

Ellis,  G.     Modern  Technical  Drawing 8vo,  *2  oo 

Ennis,  Wm.  D.     Linseed  Oil  and  Other  Seed  Oils   8vo,  *4  oo 

—  Applied  Thermodynamics 8 vo,  *4  50 

Flying  Machines  To-day 1 2mo,  *i  50 


D.  VAN  NOSTKAND  COMPANY'S  SHORT-TITLE  CATALOG     13 

Vapors  for  Heat  Engines I2mo,  *i  oo 

Erfurt,  J.    Dyeing  of  Paper  Pulp.    Trans,  by  J.  Hubner.  .8vo. 

Ermen,  W.  F.  A.     Materials  Used  in  Sizing i2mo,  *2  oo 

Evans,  C.  A.     Macadamized  Roads (In  Press.} 

Ewing,  A.  J.     Magnetic  Induction  in  Iron 8vo,  *4  oo 

Fairie,  J.     Notes  on  Lead  Ores I2mo,  *i  oo 

—  Notes  on  Pottery  Clays i2mo,  *i  50 

Fairley,  W.,  and  Andre,  Geo.  J.     Ventilation  of  Coal  Mines. 

(Science  Series  No.  58.) i6mo,  o  50 

Fairweather,  W.  C.     Foreign  and  Colonial  Patent  Laws  . .  .8vo,  *3  oo 

Fanning,  T.  T.     Hydraulic  and  Water-supply  Engineering. 8vo,  *$  oo 

Fay,  I.  W.     The  Coal-tar  Colors 8vo,  *4  oo 

Fernbach,  R.  L.    Glue  and  Gelatine 8vo,  *3  oo 

Chemical  Aspects  of  Silk  Manufacture i2mo,  *i  oo 

Fischer,  E.     The  Preparation  of  Organic  Compounds.     Trans. 

by  R.  V.  Stanford 1 2mo,  *i  25 

Fish,  J.  C.  L.     Lettering  of  Working  Drawings Oblong  80,  i  oo 

Fisher,  H.  K.  C.,  and  Darby,  W.  C.     Submarine  Cable  Testing. 

8vo,  *3  50 
Fleischmann,  W.     The  Book  of  the  Dairy.     Trans,  by  C.  M. 

Aikman 8vo,  4  oo 

Fleming,    J.    A.     The    Alternate-current    Transformer.     Two 

Volumes 8vo, 

Vol.    I.     The  Induction  of  Electric  Currents *5  oo 

Vol.  II.     The  Utilization  of  Induced  Currents *5  oo 

Propagation  of  Electric  Currents 8vo,  *s  oo 

A  Handbook  for  the  Electrical  Laboratory  and  Testing 

Room.     Two  Volumes 8vo,  each,  *5  oo 

Fleury,  P.     White  Zinc  Paints i2mo,  *2  50 

Flynn,  P.  J.    Flow  of  Water.     (Science  Series  No.  84.)  .i6mo,  o  50 

--Hydraulic  Tables.    (Science  Series  No.  66.) i6mo,  050 

Forgie,  J.    Shield  Tunneling 8vo.  (In  Press.) 

Foster,  H.  A.     Electrical  Engineers'  Pocket-book.     (Seventh 

Edition.) i2mo,  leather,  5  oo 

Engineering  Valuation  of  Public  Utilities 8vo,  *3  oo 

Handbook  of  Electrical  Cost  Data 8vo.    (In  Press) 

Fowle,  F.  F.     Overhead  Transmission  Line  Crossings  .. .  .i2mo,  *i  50 
The  Solution  of  Alternating  Current  Problems 8vo  (In  Press.) 


14     D.  VAX  NOSTRAXD  COMPANY'S  SHORT-TITLE  CATALOG 

FOA,  W.  G.     Transition  Curves.     (Science  Series  No.  no.). i6mo,  o  50 
Fox,  W.,  and  Thomas,  C.  W.     Practical  Course  in  Mechanical 

Drawing i2mo,  i  25 

Foye,  J.  C.     Chemical  Problems.     (Science  Series  No.  6r>.).i6mo,  o  50 
—  Handbook    of    Mineralogy.      (Science     Series    No.   86.). 

i6mo,  o  50 

Francis,  J.  B.     Lowell  Hydraulic  Experiments 4to,  15  oo 

Franzen,  H.     Exercises  in  Gas  Analysis i2mo,  *i  oo 

French,  J.  W.     Machine  Tools.     2  vols 4to,  *is  oo 

Freudemacher,   P.   W.    Electrical   Mining   Installations.     (In- 
stallation Manuals  Series.) i2mo,  *i  oo 

Frith,  J.     Alternating  Current  Design 8vo,  *2  oo 

Fritsch,  J.     Manufacture  of  Chemical  Manures.     Trans,  by 

D.  Grant 8vo,  *4  oo 

Frye,  A.  I.    Civil  Engineers'  Pocket-book i2mo,  leather,  *5  oo 

Fuller,  G.  W.     Investigations  into  the  Purification  of  the  Ohio 

River 4to,  *io  oo 

Furnell,  J.     Paints,  Colors,  Oils,  and  Varnishes 8vo,  *i  oo 

Gairdner,  J.  W.  I.    Earthwork 8vo  (In  Press.) 

Gant,  L.  W.     Elements  of  Electric  Traction 8vo,  *2  50 

Garcia,  A.  J.  R.  V.      Spanish-English  Railway  Terms.  . .  .8vo,  *4  50 
Garforth,  W.  E.     Rules  for  Recovering  Coal  Mines  after  Explo- 
sions and  Fires i2mo,  leather,  i  50 

Garrard,  C.  C.    Electric  Switch  and  Controlling  Gear. ...(/»  Press.) 

Gaudard,  J.     Foundations.     (Science  Series  No.  34.) i6mo,  o  50 

Gear,  H.  B.,  and  Williams,  P.  F.     Electric  Central  Station  Dis- 
tributing Systems I2mo,  *3  oo 

Geerligs,  H.  C.  P.     Cane  Sugar  and  Its  Manufacture 8vo,  *5  oo 

Geikie,  J.     Structural  and  Field  Geology 8vo,  *4  oo 

Mountains,  Their  Origin,  Growth  and  Decay 8vo,  *4  oo 

The  Antiquity  of  Man  in  Europe 8vo,  *s  oo 

Georgi,  F.,  and  Schubert,  A.     Sheet  Metal  Working.     Trans. 

by  C.  Salter 8vo,      3  oo 

Gerber,  N.     Analysis  of  Milk,   Condensed  Milk,  and  Infants' 

Milk-Food 8vo,  i  25 

Gerhard,  W.  F.     Sanitation,  Water-supply  and  Sewage  Disposal 

of  Country  Houses i2mo,  *2  oo 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITIAN  CATALOG  15 

—  Gas  Lighting.     (Science  Series  No.  in.) i6mo,  o  50 

Gerhard,  W.  P.     Household  Wastes.     (Science  Series  No.  97.) 

i6mo,  o  50 

House  Drainage.     (Science  No.  63.) i6mo,  o  50 

Sanitary  Drainage  of  Buildings.     (Science  Series  No.  93.) 

i6mo,  o  50 

Gerhardi,  C.  W.  H.     Electricity  Meters 8vo,  *4  oo 

Geschwind,  L.     Manufacture  of  Alum  and  Sulphates.     Trans. 

by  C.  Salter 8vo,  *s  oo 

Gibbs,  W.  E.     Lighting  by  Acetylene i2mo,  *i  50 

Gibson,  A.  H.     Hydraulics  and  Its  Application 8vo,  *5  oo 

Water  Hammer  in  Hydraulic  Pipe  Lines i2mo,  *2  oo 

Gibson,  A.  H.,  and  Ritchie,  E.  V.    Circular  Arc  Bow  Girder. 4to,  *s  50 

Gilbreth,  F.  B.     Motion  Study.     A  Method  for  Increasing  the 

Efficiency  of  the  Workman i2mo,  *2  oo 

—  Primer  of  Scientific  Management i2mo,  *i  oo 

Gillmore,  Gen.  Q.  A.    Limes,  Hydraulics  Cement  and  Mortars. 

«.—                                                                                             8vo,  4  oo 

Roads,  Streets,  and  Pavements i2mo,  2  oo 

Golding,  H.  A      The  Theta-Phi  Diagram i2mo,  *i  25 

Goldschmidt,  R.     Alternating  Current  Commutator  Motor .  8vo,  *3  oo 

Goodchild,  W.     Precious  Stones.     (Westminster  Series. ).8vo,  *2  oo 

Goodeve,  T.  M.     Textbook  on  the  Steam-engine i2mo,  2  oo 

Gore,  G.     Electrolytic  Separation  of  Metals 8vo,  *3  50 

Gould,  E.  S.     Arithmetic  of  the  Steam-engine i2mo,  i  oo 

Calculus.     (Science  Series  No.  112.) i6mo,  o  50 

High  Masonry  Dams.     (Science  Series  No.  22.) . .  .  i6mo,  o  50 

Practical  Hydrostatics  and  Hydrostatic  Formulas.     (Science 

Series.) i6mo,  o  50 

Gratacap,  L.  P.     A  Popular  Guide  to  Minerals 8vo,  *3  oo 

Gray,  J.     Electrical  Influence  Machines i2mo,  2  oo 

Gray,  J.     Marine  Boiler  Design i2mo,  *i  25 

Greenhill,  G.     Dynamics  of  Mechanical  Flight.  :t:il. 8vo,  *2  50 

Greenwood,  E.     Classified  Guide  to  Technical  and  Commercial 

Books 8vo,  *3  oo 

Gregorius,  R.     Mineral  Waxes.     Trans,  by  C.  Salter.  .  .i2mo,  *3  oo 

Griffiths,  A.  B.     A  Treatise  on  Manures i2mo,  3  oo 


16    D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Griffiths,  A.  B.     Dental  Metallurgy 8vo,  *3  50 

Gross,  E.     Hops 8vo,  *4  50 

Grossman,  J.     Ammonia  and  its  Compounds i2mo,  *i  25 

Groth,  L.  A.     Welding  and  Cutting  Metals  by  Gases  or  Electric- 
ity.    (Westminster  Series.) 8vo,  *2  oo 

Grover,  F.     Modern  Gas  and  Oil  Engines 8vo,  *2  oo 

Gruner,  A.     Power-loom  Weaving 8vo,  *3  oo 

Giildner,    Hugo.      Internal-Combustion    Engines.      Trans,    by 

H.  Diedrichs 410,  *io  oo 

Gunther,  C.  0.    Integration i2mo. 

Gurden,  R.  L.    Traverse  Tables folio,  half  mor.,  *7  50 

Guy,  A.  E.     Experiments  on  the  Flexure  of  Beams 8vo,  *i  25 

Haenig,  A.     Emery  and  the  Emery  Industry i2mo,  *2  50 

Hainbach,  R.     Pottery  Decoration.     Trans,  by  C.  Slater.  .  i2mo,  *3  oo 

Hale,  W.  J.     Calculations  of  General  Chemistry i2mo,  *i  oo 

Hall,  C.  H.     Chemistry  of  Paints  and  Paint  Vehicles i2mo,  *2  oo 

Hall,  G.  L.    Elementary  Theory  of  Alternate  Current  Work- 
ing  8vo,  *i  50 

Hall,  R.  H.     Governors  and  Governing  Mechanism i2mo,  *2  oo 

Hall,  W.  S.     Elements  of  the  Differential  and  Integral  Calculus 

8vo,  *2  25 

Descriptive  Geometry 8vo  volume  and  4to  atlas,  *3  50 

Haller,  G.  F.,  and  Cunningham,  E.  T.    The  Tesla  Coil i2mo,  *i  25 

Halsey,  F.  A.     Slide  Valve  Gears i2mo,  i  50 

The  Use  of  the  Slide  Rule.   (Science  Series.) i6mo,  o  50 

Worm  and  Spiral  Gearing.     (Science  Series.) i6mo,  o  50 

Hancock,  H.     Textbook  of  Mechanics  and  Hydrostatics 8vo,  i  50 

Hancock,  W.  C.  Refractory  Materials.  (Metallurgy  Series. (/;;  Press.) 

Hardy,  E.     Elementary  Principles  of  Graphic  Statics i2mo,  *i  50 

Harrison,  W.  B.     The  Mechanics'  Tool-book i2mo,  i  50 

Hart,  J.  W.     External  Plumbing  Work 8vo,  *3  oo 

Hints  to  Plumbers  on  Joint  Wiping 8vo,  *3  oo 

Principles  of  Hot  Water  Supply 8vo,  *3  oo 

• Sanitary  Plumbing  and  Drainage 8vo,  *3  oo 

Haskins,  C.  H.     The  Galvanometer  and  Its  Uses i6mo,  i  50 

Hatt,  J.  A.  H.    The  Colorist .  Second  Edition square  i2mo,  *i  50 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     17 

Hausbrand,  E.     Drying  by  Means  of  Air  and  Steam.     Trans. 

by  A.  C.  Wright i2mo,     *2  oo 

—  Evaporating,  Condensing  and  Cooling  Apparatus.     Trans. 

by  A.  C.  Wright 8vo,     *5  oo 

Hausmann,   E.     Telegraph   Engineering 8vo,    *s  oo 

Hausner,  A.     Manufacture  of  Preserved  Foods  and  Sweetmeats. 

Trans,  by  A.  Morris  and  H.  Robson 8vo,     *3  oo 

Hawkesworth,  T.     Graphical  Handbook  for  Reinforced  Concrete 

Design 4to,     *2  50 

Hay,  A.     Continuous  Current   Engineering 8vo,     *2  50 

Hayes,  H.  V.    Public  Utilities,  Their  Cost  New  and  Deprecia- 
tion   8vo,    *2  oo 

—  Public  Utilities,  Their  Fair  Present  Value  and  Return, 

8vo,  *2  oo 

Heather,  H.  J.  S.     Electrical  Engineering 8vo,  *3  50 

Heaviside,  O.     Electromagnetic  Theory.     Three  volumes. 

8vo,  Vols.  I  and  II,  each,  *s  oo 

Vol.  Ill,  *7  50 

Heck,  R.  C.  H.     Steam  Engine  and  Turbine 8vo,  *3  50 

Steam-Engine  and  Other  Steam  Motors.    Two  Volumes. 

Vol.    I.     Thermodynamics  and  the  Mechanics 8vo,  *3  50 

Vol.  II.     Form,  Construction  and  Working 8vo,  *s  oo 

—  Notes  on  Elementary  Kinematics 8vo,  boards,  *i  oo 

Graphics  of  Machine  Forces 8vo,  boards,  *i  oo 

Heermann,  P.     Dyers'    Materials.     Trans,    by   A.  C.  Wright. 

i2mo,  *2  50 
Hellot.  Macquer  and  D'Apligny.     Art  of  Dyeing  Wool,  Silk  and 

Cotton 8vo,  *2  oo 

Henrici,  O.     Skeleton  Structures 8vo,  i  50 

Hering,  D.  W.    Essentials  of  Physics  for  College  Students. 

8vo,  *i  75 
Hermann,  G.     The  Graphical  Statics  of  Mechanism.     Trans. 

by  A.  P.  Smith i2mo,  2  oo 

Herring-Shaw,  A.    Domestic  Sanitation  and  Plumbing.  Two 

Parts 8vo,  *s  oo 

Elementary  Science  of  Sanitation  and  Plumbing ....  8vo,  *2  oo 

Herzfeld,  J.     Testing  of  Yarns  and  Textile  Fabrics 8vo,  *3  $o 


18     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Hildebrandt,  A.     Airships,  Past  and  Present 8vo,  *3  50 

Hildenbrand,  B.  W.     Cable-Making.      (Science  Series  No.  32.) 

i6mo,  o  05 

Hildich,  H.     Concise  History  of  Chemistry iimo,  *i 

Hill,  J.  W.     The  Purification  of  Public  Water  Supplies.     New  52 

Edition (In  Press.) 

—  Interpretation  of  Water  Analysis (In  Press.} 

Hill,  M.  J.  M.    The  Theory  of  Proportion 8vo,  *2  50 

Hiroi,  I.     Plate  Girder  Construction.     (Science  Series  No.  95.) 

i6mo,  o  50 

Statically-Indeterminate  Stresses i2mo,  *2  oo 

Hirshfeld,    C.    F.      Engineering     Thermodynamics.     (Science 

Series.) i6mo,  o  50 

Hobart,  H.  M.     Heavy  Electrical  Engineering 8vo,  *4  50 

Design  of  Static  Transformers 8vo,  *2  oo 

Electricity 8vo,  *2  oo 

Electric  Trains 8vo,  *2  50 

Electric  Propulsion  of  Ships 8vo,  *2  oo 

Hobart,  J.  F.    Hard  Soldering,  Soft  Soldering,  and  Brazing . 

i2mo,  *i  oo 
Hobbs,  W.  R.  P.     The  Arithmetic  of  Electrical  Measurements 

i2mo,  o  50 

Hofif,  J.  N.     Paint  and  Varnish  Facts  and  Formulas i2mo,  *i  50 

Hole,  W.     The  Distribution  of  Gas 8vo,  *7  50 

Holley,  A.  L.     Railway  Practice folio,  6  oo 

Hopkins,  N.  M.     Experimental  Electrochemistry 8vo, 

Model  Engines  and  Small  Boats i2mo,  i  25 

Hopkinson,  J.,  Shoolbred,  J.  N.,  and  Day,  R.  E.     Dynamic 

Electricity.     (Science  Series  No.  71.) i6mo,  o  50 

Homer,   J.     Practical   Ironf ounding 8vo,  *2  oo 

Gear  Cutting,  in  Theory  and  Practice 8vo,  *3  oo 

Houghton,  C.  E.     The  Elements  of  Mechanics  of  Materials.  i2mo,  *2  oo 

Houllevigue,  L.     The  Evolution  of  the  Sciences 8vo,  *2  oo 

Houstoun,  R.  A.     Studies  in  Light  Production i2mo,  *2 .00 

Hovenden,  F.     Practical  Mathematics  for  Young  Engineers, 

i2mo,  *i  oo 

Howe,  G.     Mathematics  for  the  Practical  Man. i2mo,  *i  25 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     19 

Howorth,  J.     Repairing  and  Riveting  Glass,  China  and  Earthen- 
ware   8vo,  paper,  *o  50 

Hubbard,  E.     The  Utilization  of  Wood-waste 8vo,  *2  50 

Hubner,  J.    Bleaching  and  Dyeing  of  Vegetable  and  Fibrous 

Materials.     (Outlines  of  Industrial  Chemistry.)  ....  *s  oo 
Hudson,    O.    F.    Iron    and    Steel.     (Outlines    of    Industrial 

Chemistry.)    8vo,  *2  oo 

Humphrey,  J.  C.  W.    Metallography  of  Strain.     (Metallurgy 

Series)    (In  Press.) 

Humphreys,    A.    C.     The    Business    Features    of   Engineering 

Practice 8vo,  *2  50 

Hunter,  A.    Bridge  Work 8vo    (In  Press.) 

Hurst,  G.  H.     Handbook  of  the  Theory  of  Color 8vo,  *2  50 

—  Dictionary  of  Chemicals  and  Raw  Products 8vo,  *3  oo 

Lubricating  Oils,  Fats  and  Greases 8vo,  *4  oo 

Soaps 8vo,  *5  oo 

Hurst,  G.  H.,  and  Simmons,  W.  H.     Textile  Soaps  and  Oils, 

8vo,  *2  50 

Hurst,  H.  E.,  and  Lattey,  R.  T.     Text-book  of  Physics 8vo,  *3  oo 

Also  published  in  Three  Parts : 

Vol.     I.  Dynamics    and    Heat 8vo,  *i  25 

Vol.     II.  Sound  and  Light 8vo,  *i  25 

Vol.  III.  Magnetism  and  Electricity 8vo,  *i  50 

Hutchinson,  R.  W.,  Jr.     Long  Distance  Electric  Power  Trans- 
mission  i2mo,  *3  oo 

Hutchinson,  R.  W.,  Jr.,  and  Thomas,  W.  A.     Electricity  in 

Mining  i2mo, 

Hutchinson,  W.  B.     Patents  and  How  to  Make  Money  Out  of 

Them i2mo,  i  25 

Button,  W.  S.     Steam-boiler  Construction 8vo,  6  oo 

Button,  W.  S.     The  Works'  Manager's  Handbook 8vo,  6  oo 

Byde,  E.  W.     Skew  Arches.     (Science  Series  No.  15.)..  ..  i6mo,  o  50 

Hyde,  F.  S.    Solvents,  Oils,  Gums  and  Waxes i2mo,  *2  oo 

Induction  Coils.     (Science  Series  No.  53.) i6mo,  o  50 

Ingham,  A.  E.    Gearing.    A  practical  treatise 8vo,  *2  50 

Ingle,  H.     Manual  of  Agricultural  Chemistry.  8vo,  *3  oo 


20     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Innes,  C.  H.     Problems  in  Machine  Design i2ino,  *2  oo 

—  Air  Compressors  and  Blowing  Engines .  i2mo,  *2  oo 

—  Centrifugal  Pumps i2mo,  *2  GO 

-  The  Fan i2mo,  *2  oo 

Ivatts,  E.  B.     Railway  Management  at  Stations 8vo,  *2  50 

Jacob,  A.,  and  Gould,  E.  S.     On  the  Designing  and  Construction 

of  Storage  Reservoirs.     (Science  Series  No.  6.).  .i6mo,  o  50 
Jannettaz,  E.     Guide  to  the  Determination  of  Rocks.     Trans. 

by  G.  W.  Plympton I2mo,  i  50 

Jehl,  F.     Manufacture  of  Carbons 8vo,  *4  oo 

Tennings,    A.    S.     Commercial   Paints   and   Painting.     (West- 
minster Series.)    8vo,  *2  oo 

Jennison,  F.  H.     The  Manufacture  of  Lake  Pigments 8vo,  *3  oo 

Jepson,  G.     Cams  and  the  Principles  of  their  Construction..  .8  vo,  *i  50 

—  Mechanical  Drawing 8vo  (In  Preparation.) 

Jervis-Smith,  F.  J.     Dynamometers 8vo,  *3  50 

Jockin,  W.     Arithmetic  of  the  Gold  and  Silversmith i2mo,  *i  oo 

Johnson,  J.  H.    Arc  Lamps.     (Installation  Manuals  Series.) 

i2mo,  *o  75 
Johnson,    T.    M.      Ship   Wiring    and    Fitting.      (Installation 

Manuals  Series) i6mo,  *o  75 

Johnson,  W.  McA.  The  Metallurgy  of  Nickel (In  Preparation.) 

Johnston,  J.  F.  W.,  and  Cameron,  C.  Elements  of  Agricultural 

Chemistry  and  Geology i2mo,  2  60 

Joly,  J.  Radioactivity  and  Geology I2mo,  *3  oo 

Jones,  H.  C.  Electrical  Nature  of  Matter  and  Radioactivity 

i2mo,  *2  oo 

— —  New  Era  in  Chemistry i2mo,  *2  oo 

Jones,  J.  H.     Tinplate  Industry 8vo,  *3  oo 

Jones,  M.  W.     Testing  Raw  Materials  Used  in  Paint i2mo,  *2  oo 

Jordan,  L.  C.     Practical  Railway  Spiral i2mo,  Leather,  *i  50 

Joynson,  F.  H.     Designing  and  Construction  of  Machine  Gear- 
ing  8vo,  2  oo 

Jiiptner,  H.  F.  V.     Siderology:  The  Science  of  Iron 8vo,  *5  oo 

Kapp,  G.     Alternate  Current  Machinery.     (Science  Series  No. 

g6.) i6mo,  o  50 


T).  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG   21 

Keim,  A.  W.     Prevention  of  Dampness  in  Buildings  .  :  .  .     8vo,  *2  oo 
Keller,  S.  S.     Mathematics  for  Engineering  Students. 

i2mo,  half  leather, 

—  Algebra  and  Trigonometry,  with  a  Chapter  on  Vectors. ...  *i  75 

—  Plane  and  Solid  Geometry. *i  25 

and  Knox,  W.  F.     Analytical  Geometry  and  Calculus..  *2  oo 

Kelsey,   W.    R.       Continuous-current    Dynamos  and  Motors. 

8vo,  *2  50 
Kemble,  W.  T.,  and  Underbill,  C.  R.     The  Periodic  Law  and  the 

Hydrogen   Spectrum 8vo,  paper,  *o  50 

Kemp,  J.  F.     Handbook  of  Rocks. 8vo,  *i  50 

Kennedy,   A.    B.   W.,   and   Thurston,   R.   H.     Kinematics   of 

Machinery.     (Science  Series  No.  54.) i6mo,  o  50 

Kennedy,  A.  B.  W.,  Unwin,  W.  C.,  and  Idell,  F.  E.     Compressed 

Air.     (Science  Series  No.  106.) i6mo,  o  50 

Kennedy,   R.     Modern  Engines  and   Power   Generators.     Six 

Volumes 4to,  15  oo 

Single  Volumes each,  3  oo 

Electrical  Installations.     Five  Volumes 4to,  15  oo 

Single  Volumes each,  3  50 

Principles  of  Aeroplane  Construction iimo,  *i  50 

—  Flying  Machines;  Practice  and  Design i2mo,  *2  oo 

Kennelly,  A.  E.     Electro-dynamic  Machinery 8vo,  i  50 

Kent,  W.    Strength  of  Materials.     (Science  Series  No.  41.).! 6mo,  050 

Kershaw,  J.  B.  C.     Fuel,  Water  and  Gas  Analysis 8vo,  *2  50 

Electrometallurgy.     (Westminster  Series.) 8vo,  *2  oo 

—  The  Electric  Furnace  in  Iron  and  Steel  Production.. i2mo,  *i  50 
Electro-Thermal  Methods  of  Iron  and  Steel  Production, 

8vo,  *3  oo 

Kinzbrunner,  C.     Alternate  Current  Windings 8vo,  *i  50 

Continuous  Current  Armatures 8vo,  *i  50 

—  Testing  of  Alternating  Current  Machines 8vo,  *2  oc 

Kirkaldy,    W.    G.     David    Kirkaldy's    System    of    Mechanical 

Testing 4to,  10  oo 

Kirkbride,  J.     Engraving  for  Illustration 8vo,  *i  50 

Kirkwood,  J.  P.     Filtration  of  River  Waters 410,  7  50 

Kirschke,  A.    Gas  and  Oil  Engines. .    , i2mo.  *i  25 


S2     D.  VAN  NOSTRAND  COMPANY *S  SHORT-TITLE  CATALOG 

Klein,  J.  F.     Design  of  a  High  speed  Steam-engine 8vo,  *5  oo 

Physical  Significance  of  Entropy 8vo,  *i  50 

Knight,  R.-Adm.  A.  M.     Modern  Seamanship 8ro,  *7  5<> 

HalfMor.  *g  oo 

Knott,  C.  G.,  and  Mackay,  J.  S.     Practical  Mathematics.  .  .8vo,  2  oo 

Knox,  J.    Physico-chemical  Calculations i2mo,  *i  oo 

Fixation    of   Atmospheric    Nitrogen.      (Chemical   Mono- 
graphs.)      i2mo,  o  75 

Koester,  F.     Steam-Electric  Power  Pbints 4to,  *5  oo 

—  Hydroelectric  Developments  and  Engineering 4to,  *5  oo 

Koller,  T.    The  Utilization  of  Waste  Products 8vo,  *s  oo 

—  Cosmetics 8vo,  *2  50 

Kremann,  R.     Application   of   Phy.sico   Chemical   Theory  to 

Technical    Processes    and    Manufacturing    Methods. 

Trans,  by  H.  E.  Potts , 8vo,  *s  oo 

Kretchmar,  K.    Yam  and  Warp  Sizing 8vo,  *4  oo 

Lallier,  E.  V.    Elementary  Manual  of  the  Steam  Engine. 

I2mOy  *2    00 

Lambert,  T.     Lead  and  its  Compounds 8vo,  *3  50 

—  Bone  Products  and  Manures 8vo,  *3  oo 

Lamborn,  L.  L.     Cottonseed  Products 8vo,  *3  oo 

Modern  Soaps,  Candles,  and  Glycerin 8vo,  *7  50 

Lamprecht,  R.     Recovery  Work  After  Pit  Fires.      Trans,  by 

C.  Salter 8vo,  *4  oo 

Lancaster,  M.    Electric  Cooking,  Heating  and  Cleaning.  .8vo,  *i  50 
Lanchester,  F.  W.    Aerial  Flight.    Two  Volumes.    8vo. 

Vol.   I.     Aerodynamics   *6  oo 

Vol.  II.    Aerodonetics *6  oo 

Lamer,  E.  T.    Principles  of  Alternating  Currents i2mo,  *i  25 

La  Rue,  B.  F.     Swing  Bridges.     (Science  Series  No.  107.).  i6mo,  050 
Lassar-Cohn,  Dr.     Modern  Scientific  Chemistry.     Trans,  by  M. 

M.  Pattison  Muir i2mo,  *2  oo 

Latimer,  L.  H.,  Field,  C.  J.,  and  Howell,  J.  W.     Incandescent 

Electric  Lighting.     (Science  Series  No.  57.) i6mo,  o  50 

Latta,  M.  N.     Handbook  of  American  Gas-Engineering  Practice. 

8vo,  *4  50 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     23 

American  Producer  Gas  Practice 4to,  *6  oo 

Laws,  B.  C.  Stability  and  Equilibrium  of  Floating  Bodies.Svo,  *3  50 
Lawson,  W.  R.     British  Railways,  a  Financial  and  Commer- 
cial Survey ,8vo,  *2  oo 

Leask,  A.  R.     Breakdowns  at  Sea i2mo,  2  oo 

—  Refrigerating  Machinery i2mo,  2  oo 

Lecky,  S.  T.  S.     "  Wrinkles  "  in  Practical  Navigation 8vo,  *8  oo 

Le  Doux,  M.     Ice  -Making  Machines.     (Science  Series  No.  46.) 

i6mo,  o  50 
Leeds,  C.  C.    Mechanical  Drawing  for  Trade  Schools .  oblong,  4to,                 \ 

High  School  Edition *i  25    - 

Machinery  Trades  Edition *2  oo 

Lefe*vre,  L.     Architectural  Pottery.     Trans,  by  H.  K.  Bird  and 

W.  M.  Binns 4to,  *7  50 

Lehner,  S.     Ink  Manufacture.     Trans,  by  A.  Morris  and  H. 

Robson 8vo,  *2  50 

Lemstrom,  S.     Electricity  in  Agriculture  and  Horticulture.  .8 vo,  *i  50 

Letts,  E.  A.     Fundamental  Problems  in  Chemistry. .  .i2mo,  *2  oo 
Le  Van,  W.  B.    Steam-Engine  Indicator.    (Science  Series  No. 

78.) i6mo,       o  50 

Lewes,  V.  B.    Liquid  and  Gaseous  Fuels.   (Westminster  Series.) 

8vo,     *2  oo 

Carbonisation  of  Coal 8vo,    *3  oo 

Lewis,  L.  P.     Railway  Signal  Engineering 8vo,    *3  50 

Lieber,  B.  F.    Lieber's  Standard  Telegraphic  Code 8vo,  *io  oo 

Code.     German   Edition 8vo,  *io  oo 

Spanish  Edition '. 8vo,  *io  oo 

French    Edition    8vo,  *io  oo 

Terminal   Index    8vo,    *2  50 

Lieber's  Appendix    folio,  *is  oo 

Handy  Tables   4to,    *2  50 

Bankers    and    Stockbrokers'    Code    and    Merchants    and 

Shippers'  Blank  Tables  8vo,  *is  oo 

Lieber,  B.  F.     100,000,000  Combination  Code 8vo,  *io  oo 

Engineering  Code 8vo,  *i2  50 

Livermore,  V.  P.,  and  Williams,  J.     How  to  Become  a  Com- 
petent Motorman I2mo,     *i  oo 


24     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Livingstone,  R.     Design  and  Construction  of  Commutators.Svo,  *2  25 

—  Mechanical  Design  and  Construction  of  Generators.  ..8vo,  *s  50 

Lobben,  P.     Machinists'  and  Draftsmen's  Handbook 8vo,  2  50 

Lockwood,  T.  D.     Electricity,  Magnetism,  and  Electro-teleg- 
raphy  8vo,  2  50 

-  Electrical  Measurement  and  the  Galvanometer ....  i2mo,  o  75 

Lodge,  0.  J.     Elementary  Mechanics i2mo,  i  50 

—  Signalling  Across  Space  without  Wires 8vo,  *2  oo 

Loewenstein,  L.  C.,  and  Crissey,  C.  P.     Centrifugal  Pumps .  8vo,  *4  50 

Lomax,  J.  W.    Cotton  Spinning i2mo,  i  50 

Lord,  R.  T.     Decorative  and  Fancy  Fabrics 8vo,  *3  50 

Loring,  A.  E.     A  Handbook  of  the  Electromagnetic  Telegraph. 

(Science  Series  No.  39) '. i6mo,  o  50 

Low,  D.  A.  Applied  Mechanics  (Elementary) i6mo,  o  80 

Lubschez,  B.  J.  Perspective i2mo,  *i  50 

Lucke,  C.  E.  Gas  Engine  Design 8vo,  *3  oo 

Power  Plants:  their  Design,  Efficiency,  and  Power  Costs. 

2  vols (In  Preparation.) 

Lunge,  G.  Coal-tar  Ammonia.  Two  Volumes 8vo,  *i$  oo 

—  Manufacture  of  Sulphuric  Acid  and  Alkali.     Three  Volumes 

8vo, 

Vol.   I.     Sulphuric  Acid  and  Alkali.     In  three  parts. . . .  *i8  oo 
Vol.  II.     Salt  Cake,  Hydrochloric  Acid  and  Leblanc  Soda. 

In  two  parts *i5  oo 

Vol.  III.    Ammonia  Soda *io  oo 

Vol.  IV.    Electrolytic  Methods (In  Press.) 

—  Technical  Chemists'  Handbook i2mo,  leather,     *3  50 

Technical   Methods   of   Chemical   Analysis.     Trans,   by 

C.   A.  Keane.     In   collaboration   with   the   corps   of 

specialists. 

Vol.    I.     In  two  parts 8vo,  *is  oo 

Vol.  II.    In  two  parts 8vo,  *i8  oo 

Vol.  III.    In  two  parts 8vo,  *i8  oo 

The  set  complete *48  oo 

—  Technical  Gas  Analysis 8vo,    *4  oo 

Luquer,  L.  M.     Minerals  in  Rock  Sections 8vo,     *i  50 


D.  VAN  NOSTKAND  COMPANY'S  SHORT-TITLE  CATALOG    25 

Macaulay,    J.,    and    Hall,    C.      Modern    Railway    Working. 

Eight  vols 4to,  20  oc 

Each  volume  separately 3  oo 

Macewen,  H.  A.     Food  Inspection 8vo,  *2  50 

Mackenzie,  N.  F.     Notes  on  Irrigation  Works 8vo,  *2  50 

Mackie,  J.     How  to  Make  a  Woolen  Mill  Pay 8vo,  *2  oo 

Maguire,  Wm.  R.     Domestic  Sanitary  Drainage  and  Plumbing 

8vo,  4  oo 

Malcolm,  H.  W.     Submarine  Telegraph  Cable (In  Press.) 

Mallet,    A.     Compound    Engines.     Trans,    by    R.    R.    Buel. 

(Science  Series  No.  10.) i6mo, 

Mansfield,  A.  N.     Electro-magnets.     (Science  Series  No.  64) 

N*M                                                                                             i6mo,  o  50 

Marks,  E.  C.  R.     Construction  of  Cranes  and  Lifting  Machinery 

I2mo,  *i  50 

Construction  and  Working  of  Pumps i2mo,  *i  50 

Manufacture  of  Iron  and  Steel  Tubes I2mo,  *2  oo 

Mechanical  Engineering  Materials i2mo,  *i  oo 

Marks,  G.  C.     Hydraulic  Power  Engineering 8vo,  3  50 

Inventions,  Patents  and  Designs i2mo,  *i  oo 

Marlow,  T.  G.     Drying  Machinery  and  Practice 8vo,  *5  oo 

Marsh,  C.  F.     Concise  Treatise  on  Reinforced  Concrete.. .  .8vo,  *2  50 

Marsh,  C.  F.     Reinforced    Concrete    Compression    Member 

Diagram i  50 

Marsh,  C.  F.,  and  Dunn,  W.    Manual  of  Reinforced  Concrete 

and  Concrete  Block  Construction i6mo,  mor.,  *2  50 

Marshall,  W.  J.,  and  Sankey,  H.  R.    Gas  Engines.    (Westminster 

Series.) 8vo,  *2  oo 

Martin,  G.    Triumphs  and  Wonders  of  Modern  Chemistry. 

8vo,  *2  oo 

Martin,  N.    Reinforced  Concrete 8vo,  *2  50 

Martin,  W.  D.    Hints  to  Engineers i2mo,  *i  oo 

Massie,  W.  W.,  and  Underbill,  C.  R.    Wireless  Telegraphy  and 

Telephony I2nio,  *i  oo 

Mathot,  R.  E.     Internal  Combustion  Engines 8vo,  *6  oo 

Maurice,  W.     Electric  Blasting  Apparatus  and  Explosives  ..8vo,  *3  50 
Shot  Firer's  Guide 8vo,  *i  50 


26     D.  VAX  NOSTRAXD  COMPANY'S  SHORT  TlTLP:  CATALOG 

Maxwell,  J.  C.     Matter  and  Motion.     (Science  Series  No.  36.) 

i6mo,      o  50 
Maxwell,  W.  H.,  and  Brown,  J.  T.     Encyclopedia  of  Municipal 

and  Sanitary  Engineering 4to,  *io  oo 

McCullough,  E.     Practical  Surveying 8vo,    *2  50 

McCullough,  R.  S.     Mechanical  Theory  of  Heat .  8vo,       3  50 

McGibbon,  W.  C.    Indicator  Diagrams  for  Marine  Engineers, 

8vo,     *3  oo 

Marine  Engineers'  Drawing  Book oblong,  4to,    *2  oo 

Mclntosh,  J.  G,     Technology  of  Sugar 8vo,     *4  50 

Industrial  Alcohol 8vo,     *3  oo 

Manufacture  of  Varnishes  and  Kindred  Industries. 

Three  Volumes.     8vo. 

Vol.  I.     Oil  Crushing,  Refining  and  Boiling *3  50 

Vol.  II.     Varnish  Materials  and  Oil  Varnish  Making *4  oo 

Vol.  HI.     Spirit  Varnishes  and  Materials *4  50 

McKnight,   J.   D.,  and  Brown,   A.   W.     Marine   Multitubular 

Boilers *i  50 

McMaster,  J.  B.     Bridge  and  Tunnel  Centres.     (Science  Series 

No.  20.) i6mo,  o  50 

McMechen,  F.  L.     Tests  for  Ores,  Minerals  and  Metals.. .  i2mo,  *i  oo 

McPherson,  J.  A.     Water-works  Distribution 8vo,  2  50 

Melick,  C.  W.     Dairy  Laboratory  Guide i2mo,  *i  25 

Merck,    E.     Chemical    Reagents:    Their    Purity   and   Tests. 

Trans,  by  H.  E.  Schenck 8vo,  i  oo 

Merivale,  J.  H.     Notes  and  Formulae  for  Mining  Students, 

i2mo,  i  50 

Merritt,  Wm.  H.  Field  Testing  for  Gold  and  Silver .  i6mo,  leather,  i  50 
Meyer,  J.  G.  A.,  and  Pecker,  C.  G.     Mechanical  Drawing  and 

Machine  Design 4to,  5  oo 

Mierzinski,  S.     Waterproofing  of  Fabrics.     Trans,  by  A.  Morris 

and  H.  Robson 8vo,  *2  50 

Miller,  G.  A.     Determinants.     (Science  Series  No.  105.).  .i6mo, 

Milroy,  M.  E.  W.     Home  Lace -making i2mo,  *i  oo 

Mitchell,  C.  A.    Mineral  and  Aerated  Waters 8vo,  *3  oo 

and    Prideaux,    R.    M.      Fibres    Used    in    Textile    and 

Allied  Industries 8vo,  *3  oo 


D.  VAN  NOSTRAND  COMPANY'S  SHORT  TITLE  CATALOG      27 

Mitchell,  C.  F.  and  G.  A.     Building  Construction  and  Draw- 
ing       I2H10 

Elementary  Course,  *i  50 

Advanced  Course,  *2  50 
Monckton,  C.  C.  F.     Radiotelegraphy.     (Westminster  Series.) 

8vo,  *2  oo 
Monteverde,  R.  D.     Vest  Pocket  Glossary  of  English-Spanish, 

Spanish-English  Technical  Terms 641110,  leather,  *i  oo 

Montgomery,  J.  H.     Electric  Wiring  Specifications i2mo,  *i  oo 

Moore,  E.  C.  S.     New  Tables  for  the  Complete  Solution  of 

Ganguillet  and  Kutter's  Formula 8vo,  *5  oo 

Morecroft,  J.  H.,  and  Hehre,  F.  W.    Testing  Electrical  Ma- 
chinery   8vo,  *i  50 

Morgan,  A.  P.     Wireless  Telegraph  Construction  for  Amateurs. 

i2mo,  *i  50 

Moses,  A.  J.     The  Characters  of  Crystals 8vo,  *2  oo 

and  Parsons,  C.  L.     Elements  of  Mineralogy 8vo,  *2  50 

Moss,    S.    A.     Elements    of    Gas    Engine    Design.     (Science 

Series.) i6mo,  o  50 

The  Lay-out  of  Corliss  Valve  Gears.      (Science  Series). 

i6mo,  o  50 

Mulford,  A.  C.    Boundaries  and  Landmarks 8vo,  *i  oo 

Mullin,  J.  P.     Modern  Moulding  and  Pattern-making.  .  .  .  i2mo,  2  50 
Munby,  A.  E.     Chemistry  and  Physics  of  Building  Materials. 

(Westminster  Series.) 8vo,  *2  oo 

Murphy,  J.  G.     Practical  Mining i6mo,  i  oo 

Murphy,  W.  S.    Textile  Industries,  8  vols *2o  oo 

(Sold   separately.)    each,  *s  oo 

Murray,  J.  A.     Soils  and  Manures.     (Westminster  Series.). 8 vo,  *2  oo 

Naquet,  A.     Legal  Chemistry i2mo,  2  oo 

Nasmith,  J.     The  Student's  Cotton  Spinning 8vo,  3  oo 

Recent  Cotton  Mill  Construction i2mo,  2  oo 

Neave,  G.  B.,  and  Heilbron,  I.  M.    Identification  of  Organic 

Compounds : i2mo,  *i  25 

Neilson,  R.  M.     Aeroplane  Patents 8vo,  *2  oo 

Nerz,  F.     Searchlights.     Trans,  by  C.  Rodgers 8vo,  *3  OQ 


28      D.  VAN  NOSTRAND  COMPANY'S  SHORT  TITLE  CATALOG 

Nesbit,  A.  F.    Electricity  and  Magnetism (In  Preparation.) 

Neuberger,   H.,   and  Noalhat,  H.     Technology   of   Petroleum. 

Trans,  by  J.  G.  Mclntosh. 8vo,  *io  oo 

Newall,  J.  W.  Drawing,  Sizing  and  Cutting  Bevel-gears.  .8vo,  i  50 
Newbiging,  T.  Handbook  for  Gas  Engineers  and  Managers, 

8vo,  *6  50 

Nicol,  G.     Ship  Construction  and  Calculations 8vo,  *4  50 

Nipher,  F.  E.     Theory  of  Magnetic  Measurements i2mo,  i  oo 

Nisbet,  H.     Grammar  of  Textile  Design 8vo,  *3  oo 

Nolan,  H.     The  Telescope.     (Science  Series  No.  51.) i6mo,  o  50 

North,  H.  B.    Laboratory  Experiments  in  General  Chemistry 

i2mo,  *i  oo 

Nugent,  E.     Treatise  on  Optics i2mo,  i  50 

O'Connor,  H.  The  Gas  Engineer's  Pocketbook.  ..  i2mo,  leather,  350 
Ohm,  G.  S.,  and  Lockwood,  T.  D.  Galvanic  Circuit.  Trans,  by 

William  Francis.  (Science  Series  No.  102.).  .  .  .  i6mo,  o  50 

Olsen,  J.  C.  Text  book  of  Quantitative  Chemical  Analysis .  .8vo,  *4  oo 
Olsson,  A.  Motor  Control,  in  Turret  Turning  and  Gun  Elevating. 

(U.  S.  Navy  Electrical  Series,  No.  i.) .  ...i2mo,  paper,  *o  50 

Ormsby,  M.  T.  M.  Surveying izmo,  i  50 

Oudin,  M.  A.  Standard  Polyphase  Apparatus  and  Systems  . .  8vo,  *3  oo 

Owen,  D.  Recent  Physical  Research 8vo,  *i  50 

Pakes,  W.  C.  C.,  and  Nankivell,  A.  T.    The  Science  of  Hygiene. 

8vo,  *i  75 
Palaz,  A.     Industrial  Photometry.     Trans,  by  G.  W.  Patterson, 

Jr 8vo,  *4  oo 

Pamely,  C.     Colliery  Manager's  Handbook 8vo,  *io  oo 

Parker,  P.  A.  M.     The  Control  of  Water 8vo,    *s  oo 

Parr,  G.  D.  A.     Electrical  Engineering  Measuring  Instruments. 

8vo,  *3  So 

Parry,  E.  J.     Chemistry  of  Essential  Oils  and  Artificial  Per- 
fumes  8vo,  *5  oo 

Parry,  E.  J.     Foods  and  Drugs.    Two  Volumes 8vo. 

Vol.   I.     Chemical  and  Microscopical  Analysis  of  Food 

and  Drugs *7  •  5° 

Vol.  II.     Sale  of  Food  and  Drugs  Acts *3  oo 

and  Coste,  J.  H. 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG   29 

Parry,  L.     Notes  on  Alloys 8vo,  3  oo 

Metalliferous  Wastes    8vo,  2  oo 

Analysis  of  Ashes  and  Alloys 8vo,  2  oo 

Parry,  L.  A.     Risk  and  Dangers  of  Various  Occupations 8vo,  *3  oo 

Parshall,  H.  F.,  and  Hobart,  H.  M.     Armature  Windings  ....  4to,  *y  50 

—  Electric  Railway  Engineering 4to,  *io  oo 

Parsons,  S.  J.     Malleable  Cast  Iron 8vo,  *2  50 

Partington,  J.  R.    Higher  Mathematics  for  Chemical  Students 

i2mo,  *2  oo 

Textbook  of  Thermodynamics 8vo,  *4  oo 

Passmore,  A.  C.     Technical  Terms  Used  in  Architecture  . . .  8vo,  *3  50 

Patchell,  W.  H.     Electric  Power  in  Mines 8vo,  *4  oo 

Paterson,  G.  W.  L.    Wiring  Calculations i2mo,  *2  oo 

—  Electric  Mine  Signalling  Installations i2mo,  *i  50 

Patterson,  D.     The  Color  Printing  of  Carpet  Yarns 8vo,  *3  50 

—  Color  Matching  on  Textiles 8vo,  *3  oo 

Textile  Color  Mixing 8vo,  *3  oo 

Paulding,  C.  P.     Condensation  of  Steam  in  Covered  and  Bare 

Pipes 8vo,  *2  oo 

Transmission  of  Heat  Through  Cold-storage  Insulation 

1 2  mo,    *i  oo 

Payne,  D.  W.    Iron  Founders'  Handbook (In  Press.} 

Peddie,  R.  A.    Engineering  and  Metallurgical  Books. .  . .  i2mo,  *i  50 

Peirce,  B.     System  of  Analytic  Mechanics 4to,  10  oo 

Pendred,  V.     The  Railway  Locomotive.     (Westminster  Series.) 

8vo,  *2  oo 

Perkin,  F.  M.     Practical  Method  of  Inorganic  Chemistry . .  i2mo,  *i  oo 

and  Jaggers,  E.  M.     Elementary  Chemistry i2mo,  *i  oo 

Perrine,  F.  A.  C.     Conductors  for  Electrical  Distribution  .  . .  8vo,  *3  50 

Petit,  G.     White  Lead  and  Zinc  White  Paints 8vo,  *i  50 

Petit,  R.     How  to  Build  an  Aeroplane.     Trans,  by  T.  O'B. 

Hubbard,  and  J.  H.  Ledeboer. 8vo,  *i  50 

Pettit,  Lieut.  J.  S.     Graphic  Processes.     (Science  Series  No.  76.) 

i6mo,  o  50 
Philbrick,  P.  H.     Beams  and  Girders.     (Science  Series  No.  88.) 

i6mo, 

Phillips,  J.     Gold  Assaying 8vo,  *2  50 

Dangerous  Goods 8vo,  3  50 


30  D.  VAN  XOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Phin,  J.     Seven  Follies  of  Science i2mo,  *i  25 

Pickworth,  C.  N.     The  Indicator  Handbook.     Two  Volumes 

i2mo,  each,  i  50 

—  Logarithms  for  Beginners I2mo,  boards,  o  50 

—  The  Slide  Rule i2mo,  i  oo 

Plattner's  Manual  of    Blowpipe  Analysis.     Eighth  Edition,  re- 
vised.    Trans,  by  H.  B.  Cornwall 8vo,  *4  oo 

Plympton,  G.W.  The  Aneroid  Barometer.  (Science  Series.). i6mo,  o  50 

How  to  become  an  Engineer.     (Science  Series  No.  100.) 

i6mo,  o  50 

Van  Nostrand's  Table  Book.    (Science  Series  No.  104). 

i6mo,  o  50 
Pochet,  M.  L.     Steam  Injectors.     Translated  from  the  French. 

(Science  Series  No.  29.) i6mo,  o  50 

Pocket  Logarithms  to  Four  Places.     (Science  Series.). ....  i6mo,  o  50 

bather,  i  oo 

Polleyn,  F.    Dressings  and  Finishings  for  Textile  Fabrics. 8vo,  *3  oo 

Pope,  F.  G.    Organic  Chemistry i2mo,  *2  25 

Pope,  F.  L.     Modern  Practice  of  the  Electric  Telegraph.. .   8 vo,  i  50 

Popplewell,  W.  C.    Prevention  of  Smoke. 8vo,  *3  50 

Strength   of  Materials 8vo,  *i  75 

Porritt,  B.  D.    The  Chemistry  of  Rubber.     (Chemical  Mono- 
graphs.)      i2mo,  *o  75 

Porter,  J.  R.     Helicopter  Flying  Machines i2mo,  i  25 

Potts,  H.  E.  Chemistry  of  the  Rubber  Industry.     (Outlines  of 

Industrial  Chemistry.) 8vo,  *2  oo 

Practical  Compounding  of  Oils,  Tallow  and  Grease 8vo,  *3  50 

Pratt,  K.    Boiler  Draught i2mo,  ¥i  25 

High  Speed   Steam   Engines 8vo,  *2  oo 

Pray,  T.,  Jr.     Twenty  Years  with  the  Indicator 8vo,  2  50 

Steam  Tables  and  Engine  Constant 8vo,  2  oo 

Prelini,  C.     Earth  and  Rock  Excavation 8vo,  *3  oo 

Dredges  and  Dredging 8vo,  *3  oo 

Graphical  Determination  of  Earth  Slopes 8vo,  *2  oo 

Tunneling 8vo,  *3  oo 

Prescott,  A.  B.     Organic  Analysis 8vo,  5  oo 

and  Johnson,  0.  C.    Quantitative  Chemical  Analysis .  8vo,  *3  50 

—  and  Sullivan,  E.  C.    First  Book  in  Qualitative  Chemistry 

i2mo,  *i  50 


-    D.  VAN  NOSTRAND  COMPANY'S  SPIOIIT-TITL^  CATALOG     31 

Prideaux,  E.  B.  R.     Problems  in  Physical  Chemistry 8vo,  *2  oo 

Primrose,  G.  S.  C.     Zinc.     (Metallurgy  Series.) (In  Press.) 

Pullen,  W.  W.  F.     Application  of  Graphic  Methods  to  the  Design 

of  Structures i2mo,  *2  50 

—  Injectors:  Theory,  Construction  and  Working i2mo,  *i  50 

—  Indicator  Diagrams   8vo,  *2  50 

—  Engine   Testing    8vo,  *4  50 

Pulsifer,  W.  H.     Notes  for  a  History  of  Lead 8vo,  4  oo 

Putsch,  A.     Gas  and  Coal-dust  Firing 8vo,  *3  oo 

Pynchon,  T.  R.     Introduction  to  Chemical  Physics 8vo,  3  oo 

Rafter,  G.  W.     Mechanics  of  Ventilation.     (Science  Series  No. 

330 i6mo,  o  50 

—  Potable  Water.     (Science  Series  No.  103.) i6mo,  o  50 

—  Treatment  of  Septic  Sewage.     (Science  Series.).  .  .  .i6mo,  o  50 

—  and  Baker,  M.  N.    Sewage  Disposal  in  the  United  States 

4to,  *6  oo 

Raikes,  H.  P.     Sewage  Disposal  Works 8vo,  *4  oo 

Ramp,  H.  M.     Foundry  Practice (In  Press.) 

Randau,  P.     Enamels  and  Enamelling 8vo,  *4  co 

Rankine,  W.  J.  M.     Applied  Mechanics 8vo,  5  oo 

—  Civil  Engineering 8vo,  6  50 

—  Machinery  and  Millwork 8vo,  5  oo 

The  Steam-engine  and  Other  Prime  Movers 8vo,  5  oo 

and  Bamber,  E.  F.     A  Mechanical  Textbook 8vo,  3  50 

Raphael,  F.  C.     Localization  of    Faults  in  Electric  Light  and 

Power  Mains 8vo,  *3  oo 

Rasch,  E.    Electric  Arc  Phenomena.     Trans,  by  K.  Tornberg. 

8vo,  *2  oo 

Rathbone,  R.  L.  B.     Simple  Jewellery 8vo,  *2  oo 

Rateau,   A.     Flow  of   Steam  through   Nozzles    and    Orifices. 

Trans,  by  H.  B.  Brydon : 8vo,  *i  50 

Rautenstrauch,  W.     Notes  on  the  Elements  of  Machine  Design, 

8vo,  boards,  *i  50 

Rautenstrauch,  W.,  and  Williams,  J.  T.     Machine  Drafting  and 
Empirical  Design. 

Part    I.  Machine  Drafting 8vo,  *i  25 

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32     D.  VAN  NOSTRAND  COMPANY'S  SHORT -TITLE  CATALOG    . 

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Walker,  F.    Aerial  Navigation 8vo,  2  oo 

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Watkins,  A.    Photography,     (Westminster  Series.) 8vo,  *2  oo 

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